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Math worksheets: Convert mixed numbers to decimals (various denominators) Below are six versions of our grade 5 math worksheet on converting mixed numbers to decimals; denominators of 2, 4, 5, 10, 20, 25, 50, 100 are used. Fraction to Decimal Converter. 4 3/11 inches to decimal Here is the next length of inches that we have converted to decimal. One can think about mixed numbers (as well as decimals greater than 1) as the sum of a whole and a … In this step-by-step guide, we'll show you how to turn any fraction into a decimal … Here we will show you step-by-step with detailed explanation how to convert 14 9/32 as a decimal number. The numbers in the mixed fraction are defined as follow : 10 = whole number 6 = numerator 9 = denominator. Let’s go back to our lunch order to see how we can convert decimal numbers to fractions. A decimal with a whole part can be written as a mixed number. Since 4 is the denominator in the original fraction part we will use it as our common denominator. 29/4 = 29 ? Everything You Need to Ace Math in One Big Fat Notebook, Online Calculator | Terms | Privacy | : 1/2, 2 1/2, 5/3, etc. We often need to rewrite decimals as fractions or mixed numbers. Here are the detailed math steps we use to convert 10 1/3 mixed number to decimal form: Step 1: Multiply the whole number by the denominator: 10 × 3 = 30 Step 2: Add the product you got in Step 1 to the numerator: 30 + 1 = 31 Step 3: Divide the sum from Step 2 by the denominator: 31 ÷ 3 = 10.333333 That's it folks! Step 3: Reduce the fraction. Some of the worksheets for this concept are Mixed numbers to decimals, Mixed numbers to decimals, Converting mixed numbers to decimals, Mixed numbers to decimals, Decimals mixed numbers, Decimals work, Changing improper fractions to mixed numbers, Multiplying mixed numbers. Enter your mixed number: / First, note that 8 3/10 is a mixed number, also known as mixed fraction. Mixed number 64/90 to decimal. A - 21.10. For example, to convert 3 1/2 to a decimal, we first write the whole number followed by a decimal point. In order to convert mixed number 10 6/9 to decimal form, you follow these steps: Convert the proper fraction to decimal: Divide the numerator by the denominator. Convert ID: 906792 Language: English School subject: Math Grade/level: Grade 4 Age: 9-10 Main content: Decimals and fractions Other contents: Add to my workbooks (31) Download file pdf Embed in my website or blog For example, to convert 3 1/2 to a decimal, we first write the whole number followed by a decimal point. To convert a decimal to a fraction see the D - 21.13. If it is not zero, the decimal converts to a mixed number. To convert fractions to decimals: If the fraction is a mixed number, change it to an improper fraction. : 1/2, 2 1/2, 5/3, etc. 1. Use our fraction to decimal calculator to convert any fraction to a decimal and to know if it is a terminating or a recurring (repeating) decimal. Shows the work to change the mixed number to a percent. Convert mixed numbers or mixed fractions to decimal numbers. Step 1: Make a fraction with the decimal number as the numerator (top number) and a 1 as the denominator (bottom number). https://www.calculatorsoup.com - Online Calculators. Add this decimal number to the whole number part of the mixed number A mixed number is a whole number plus a fraction. Examples. Grade 4, 5 (821) Decimal Place Value. Fraction input: You may enter positive and negative fractions. Enter your mixed number: / First, note that 2 12/25 is a mixed number, also known as mixed fraction. The numbers in the mixed fraction are defined as follow : 8 = whole number 3 = numerator 10 = denominator. \begin {align*}\frac {13} {100}\end {align*} is 13 hundredths. Practice. I have a set of 200 decimals that I would like converted to either improper fractions or mixed numbers (preferably the latter) in batch. To change a mixed number into a decimal number, you first convert the mixed number into an improper fraction. Improve your math knowledge with free questions in "Convert mixed numbers to decimals" and thousands of other math skills. Similar: Convert decimals to fractions Convert fractions to decimals Then add the decimal number to the whole number. C - 21.12. Convert the fraction part of the mixed number to a decimal 2. \begin {align*}\frac {13} {100} = 0.13\end {align*} Then add the decimal number to the whole number. We can convert a mixed number (or mixed fraction) to a decimal. How to convert 1 23/80 to a recurring or terminating decimal. More Decimals Lesson . Grade 4, 5 (695) Decimal Number Lines. Fractions/decimals worksheets: convert 1 or 2 digit decimals to mixed numbers. 4/5 = 0.8 So, $15 \frac{5}{64} = 15.078125$ Show Answer. To enter a mixed number, place a space between the whole number and the fraction, like this: 2 1/2 For example, to convert 5 4/5, we will do the following. 1 23/80 is equal to 1.2875 in decimal form. They will convert the mixed number (all fractions have common denominators and are in hundredths) to a decimal number (to the hundredths place) and color in the squares with the answers . 1. Note that 2 1/2 means two and half = 2 + 1/2 = 2.5: Answer: Fraction to decimal explained: Detailed result here. Mixed number to Decimal ; 7 3/50 as a decimal. I would like something that will take all 4 of those decimals and convert them to either improper fractions or mixed numbers. Q 10 - Convert $\mathbf {21 \frac{3}{25}}$ into a terminating decimal. Mixed number 57/9 to decimal. Grades: 4 th, 5 th. Mixed number 71/5 to decimal. And this will be the result of the operation. It has a whole number plus a proper fraction. Mixed number to decimal calculator finds the decimal equivalent by converting a mixed number, fraction, integer or whole number to a decimal and shows the work. A mixed number is a whole number as well as some parts of a whole as fraction. 2 19/20 is equal to 2.95 in decimal form. To convert a decimal to a mixed number, leave the whole number as same and convert the decimal part. Whole number and fraction with denominator of 10, 100 or 1000 % Progress . Enter your mixed number: / First, note that 10 6/9 is a mixed number, also known as mixed fraction. © 2006 -2021CalculatorSoup® Here are some examples: The fraction 3/4: Divide 3 by 4, and you end up with the decimal .75. To change a mixed number into a decimal number, you first convert the mixed number into an improper fraction. About This Quiz & Worksheet. Convert the fraction part of the mixed number to a decimal Let’s go back to our lunch order to see how we can convert decimal numbers to fractions. Strategies to convert mixed fractions to decimals: Change the mixed fraction to an improper fraction and then divide. Note that 2 1/2 means two and half = … Mixed number to Decimal ; 14 9/32 as a decimal. Fraction to Decimal Converter. At first, converting the fractional part of mixed number into a terminating decimal $\frac{5}{64} = 0.078125$ Step 2: Keeping whole number part as it is. The decimal number obtained shall then be added, with the original whole part of the mixed fraction: 3 + 0,5= 3,5 . Converting a mixed number to a decimal is very straightforward. 2) Then, place the whole number part to the left of the decimal point. We next take the fractional part of mixed number and set it up as a long division and obtain its decimal value. A mixed number, also called a mixed fraction, has both a whole number and a fractional part. Below are six versions of our grade 6 math worksheet on converting mixed numbers to decimals. Enter a fraction value: Ex. https://www.turtlediary.com/video/converting-mixed-numbers-to-decimals.html Then, you convert the fraction into a decimal number by performing a simple division step. Step 2: Keeping whole number part as it is. Follow these 2 steps to convert a mixed number to a decimal: A mixed number is a whole number plus a fraction. : 1/2, 2 1/2, 5/3, etc. 2. Note that 2 1/2 means two and half = 2 + 1/2 = 2.5: Answer: Fraction to decimal explained: Detailed result here. Mixed number to decimal will convert any fraction or mixed fraction to a decimal quickly and eaisly. Click Create Assignment to assign this modality to your LMS. Note that this solution works even if the fraction part of the mixed number is an improper fraction. B - 21.11. Question 1 : Write the mixed number 2 5/16 as a decimal. 88.0283. Math worksheets: rewriting mixed numbers as decimals. Worksheets > Math > Grade 6 > Fractions vs. Decimals > Mixed numbers to decimals. The decimal forms of rational numbers either end or repeat a pattern. Worksheets > Math > Grade 5 > Fractions vs decimals > Mixed numbers to decimals. Alternatively you can convert a mixed number to a decimal by first converting the mixed number to two fractions, adding them and simplifying to a decimal. Converting Mixed Numbers to Decimals To write a mixed number as a decimal, 1) Convert the fractional part of the mixed number to a decimal. Mixed number to Decimal ; 10 6/9 as a decimal. 8.035. It has a whole number plus a proper fraction. Convert the following mixed expression to a decimal number: To begin with, the same part of the mixed fraction will be taken as the whole part, while the numerator will be divided by the denominator: 1 : 2 = 0, 5. : 1/2, 2 1/2, 5/3, etc. You can convert a mixed number to a decimal by rewriting the fractional part of the number as a decimal. If the division doesn’t come out evenly, round the decimal off. Mixed number to Decimal ; 8 3/10 as a decimal. Mixed number 2 19/20 to decimal Convert the fractional part only and then add it to the whole number. The fraction 61/3 is equal to 20.333333333333 when converted to a decimal. Note that 2 1/2 means two and half = 2 + 1/2 = 2.5: Answer: Fraction to decimal explained: Detailed result here. 6,989 Downloads Read Lesson! Add the result to the whole number part of the mixed number. You can also see our It has a whole number plus a proper fraction. Write the whole number. Look at the number to the left of the decimal. Answer : C Explanation. See below detalis on how to convert the fraction 64/90 to a decimal value. Enter a fraction value: Ex. You will also need to be able to reverse the conversion. The fraction 57/9 is equal to 6.3333333333333 when converted to a decimal. In decimals, the digits to the left of the decimal point represents the whole part and the digits Here we will show you step-by-step with detailed explanation how to convert 10 6/9 as a decimal number. Mixed Number to Decimal Calculator to convert fraction to decimal. First, note that 8 3/10 is a mixed number, also known as mixed fraction. Grade 4 Fractions Worksheet - Convert mixed numbers to decimals Author: K5 Learning Subject: Grade 4 Fractions Worksheet Keywords: Grade 4 Fractions Worksheet - Convert mixed numbers to decimals math practice printable elementary school Created Date: 20160125030813Z Mixed Number to Decimal. 12,136 Downloads Read Lesson! Mixed number. Next, given that you have x decimal places, multiply numerator and denominator by 10 x. It has a whole number plus a proper fraction. Similar: Writing decimals as mixed numbers Writing fractions as decimals using long division Enter a fraction value: Ex. Enter a fraction value: Ex. Write the fraction. Mixed number to decimal is a calculator to convert mixed fraction to decimal with instructions on how to cnvert a mixed number to a decimal. 1. We know that $5.03$ means $5$ dollars and $3$ cents. The whole number part in the mixed number will be the same as the whole number part in the decimal. In order to convert mixed number 8 3/10 to decimal … Follow these 2 steps to convert a mixed number to a decimal: 1. The decimal value obtained above becomes the decimal part of the decimal number. Fraction to Decimal Converter. Learn to convert given decimal values to mixed numbers. Answer : : 1/2, 2 1/2, 5/3, etc. Answer : A Explanation. Enter a fraction value: Ex. Try Another Version of This Question Question 1 Convert the decimal 7.28 7.28 into a mixed number. Decimals and Fractions - Mixed Numbers Worksheet starts off with visual models for mixed numbers to aid in understanding a decimal greater than 1 as a mixed number. For example, in $1 \frac{1}{4}, \: \frac{1}{4} = 0.25$, whole number part 1 of mixed number is retained in decimal number. 5+0.8 = 5.8, To advance your math, read Converting Mixed Fractions to Decimals : In this section, we will learn, how to convert mixed fractions to decimals. The process for converting between mixed numbers and decimals; Practice Exams. Converting a mixed number to a decimal is very straightforward. Here are the detailed math steps we use to convert 7 3/3 mixed number to decimal form: Step 1: Multiply the whole number by the denominator: 7 × 3 = 21 Step 2: Add the product you got in Step 1 to the numerator: 21 + 3 = 24 Step 3: Divide the sum from Step 2 by the denominator: 24 ÷ 3 = 8 That's it folks! Below are six versions of our grade 4 math worksheet on converting decimal amounts to mixed numbers. Definition of mixed decimal : a mixed number whose fractional part is a decimal fraction (as in 7.238) Improve your math knowledge with free questions in "Convert fractions or mixed numbers to decimals" and thousands of other math skills. Types: Worksheets, Fun Stuff, Handouts. Following is a step by step instruction on how to convert a mixed number to a decimal. Final Exam 6th-8th Grade Math: Practice & Review Status: Not Started. In Mathematics, mixed number to decimal is the conversion of the mixed fraction to the decimal number. To write a mixed number as a decimal, 1) Convert the fractional part of the mixed number to a decimal. 1: 3: 23: 2.5: 4.03: Converting mixed numbers to improper fractions. To convert fractions to decimals you just divide the top by the bottom — divide the numerator by the denominator — and if the division doesn’t come out evenly, you can stop after a certain number of decimal places and round off. It's very common when learning about fractions to want to know how convert a mixed fraction like 471 203/176 into a decimal. The fraction 11/40 is equal to 0.275 when converted to a decimal. Similar: Convert proper fractions to decimals Mixed number 91/2 to decimal. We can represent the decimal number in the fraction and vice versa. The mixed number to percentage calculator finds the decimal equivalent by finding the decimal value of the fraction, adding the decimal to the whole number part of the mixed number, and multiplying by 100 to get the percent. Mixed numbers to decimals Grade 4 Fractions Worksheet. Answers come with explanation and the underlying math. Subjects: Math, Fractions. 4 = 7.25. Mixed number to decimal will convert any fraction or mixed fraction to a decimal quickly and eaisly. For example: 1.875. Add the result to the whole number part of the mixed number. These worksheets are pdf files.. Divide the numerator by the denominator. © 2021 Online Calculator, Everything You Need to Ace Math in One Big Fat Notebook. Displaying top 8 worksheets found for - Mixed Numbers To Decimals. Determine the place value of the final digit. Then, we get 1/2 = 0.5. Step 1: At first, converting the fractional part of mixed number into a terminating decimal $\frac{3}{5} = 0.6$. 19,339 Downloads Read Lesson! Fraction to Decimal Converter. Mixed number to Decimal ; 2 12/25 as a decimal. See below detalis on how to convert the fraction 11/40 to a decimal value. Enter your mixed number: / First, note that 14 9/32 is a mixed number, also known as mixed fraction. Here we will show you step-by-step with detailed explanation how to convert 7 3/50 as a decimal number. You can use this quiz and worksheet to determine if you know how to convert decimals to mixed numbers. Reduce fractions after changing from decimal form. 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# DRYing out my Sass, looping through selectors

Need help with DRYing out my Sass... there is just so much repetition in here:

.cta_foo {      @include linear-gradient(#f69c28,#b75d00,#b75d00); text-shadow: -1px -1px 0 #b85e00;
&:hover {   @include linear-gradient(#FFAC41,#E77500,#E77500); } }
&:hover {   @include linear-gradient(#25CF39,#179F27,#179F27); } }
&:hover {   @include linear-gradient(#72B4EF,#448BCA,#448BCA); } }

I started out with an @each directive but just got confused quickly. I ask not just for this one instance but because I'm trying to wrap my head around maps and lists and loops and such.

### Update

@mixin linear-gradient just generates a linear-gradient with fallback color:

@mixin linear-gradient($start,$end,$fallback) { background-color:$fallback; background-image: linear-gradient($start,$end);
@include ms-filter($start,$end);
}

Well I finally figured out a way to do what I had in my mind:

@each $a,$b, $c,$d in (
(foo,   (#f69c28,#b75d00,#b75d00), #b85e00, (#FFAC41,#E77500,#E77500)),
(bar,   (#FFAC41,#E77500,#E77500), #008A0F, (#25CF39,#179F27,#179F27)),
(baz,   (#67a9e4,#3a7ab2,#3a7ab2), #4477a5, (#72B4EF,#448BCA,#448BCA))
) {
.foo_#{$a} { @include linear-gradient(nth($b,1),nth($b,2),nth($b,3)); text-shadow: -1px -1px 0 $c; &:hover { @include linear-gradient(nth($d,1),nth($d,2),nth($d,3)); }
}
}

But really it doesn't look like it's any clearer (likely it's worse) and it only dropped from 493 characters to 420. I guess my intention, though, was just knowing HOW to do it. Perhaps this is one case where I don't need to be DRY if it makes the code harder to understand.

• But you still repeat colors twice in each row, if you get rid of that it will be even shorter. And maybe you could compile the lighter or darker shade with colour functions instead? – ferada Feb 12 '15 at 18:03
• Well that won't always be the case (that the fallback color and end color are the same), as sometimes I'll have fallbacks that may not even use either. Stupid WCAG2.0 requires 4.5 contrast ratio and I might have a light colored background necessitating dark text and have to get creative to pass accessibility tests. Neither here nor there, but thanks for pointing that out anyways. – Phil Tune Feb 12 '15 at 19:43
• Ultimately, I don't like this solution because it makes the code even HARDER to read, and the current solution isn't terrible by itself. But it WAS an answer to what I initially was looking for. – Phil Tune Feb 12 '15 at 19:45

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## CONF notes of the ATLAS Collaboration - TOP group

Full Title Ref Code Publication Date Groups
"A search for tt resonances in lepton plus jets events with ATLAS using 14 fb-1 of proton-proton collisions at √s = 8 TeV" ATLAS-CONF-2013-052 2013/05/13 TOPQ / EXOT
"A search for tt resonances using lepton plus jets events in proton-proton collisions at √s = 8 TeV with the ATLAS detector" ATLAS-CONF-2015-009 2015/03/17 TOPQ / EXOT
"Measurement of lepton differential distributions and the top quark mass in tt production in $pp$ collisions at √s=8 TeV with the ATLAS detector" ATLAS-CONF-2017-044 2017/07/05 TOPQ
"Measurement of the tt production cross-section using eμ events with b-tagged jets in pp collisions at √s=13 TeV with the ATLAS detector" ATLAS-CONF-2016-005 2016/03/14 TOPQ
"Measurement of the ttW and ttZ production cross sections in pp collisions at √s = 8 TeV with the ATLAS detector" ATLAS-CONF-2015-032 2015/07/22 TOPQ
"Measurements of tt differential cross-sections in the all-hadronic channel with the ATLAS detector using highly boosted top quarks in pp collisions at $√s$ = 13 TeV" ATLAS-CONF-2016-100 2016/09/16 TOPQ
"Search for ttZ production in the three lepton final state with 4.7 fb-1 of √s = 7 TeV pp collision data collected by the ATLAS detector" ATLAS-CONF-2012-126 2012/08/24 TOPQ
A combined measurement of the top quark pair production cross-section using dilepton and single-lepton final states ATLAS-CONF-2011-040 2011/03/22 TOPQ
A search for Flavour Changing Neutral Currents in Top Quark Decays t → qZ at √s = 7 TeV in 0.70 fb-1 of pp collision data collected with the ATLAS Detector ATLAS-CONF-2011-154 2011/11/11 TOPQ
A Search for New High-Mass Phenomena Producing Top Quarks with the ATLAS Experiment ATLAS-CONF-2011-070 2011/05/06 TOPQ / EXOT
A Search for tt̄ Resonances in the Lepton Plus Jets Channel using 2.05 fb-1 pp Collisions at √s = 7 TeV ATLAS-CONF-2012-029 2012/03/09 TOPQ / EXOT
A search for tt̄ resonances in the lepton plus jets final state using 4.66/fb of pp collisions at √s = 7 TeV ATLAS-CONF-2012-136 2012/09/20 TOPQ / EXOT
A Search for tt Resonances in the Dilepton Channel in 1.04 fb-1 of pp Collisions at √s = 7 TeV ATLAS-CONF-2011-123 2011/08/20 EXOT / TOPQ
A Search for ttbar Resonances in the Lepton Plus Jets Channel using 200/pb of pp Collisions at √s = 7 TeV ATLAS-CONF-2011-087 2011/06/05 TOPQ / EXOT
Background studies for top-pair production in lepton plus jets final states in √s=7 TeV ATLAS data ATLAS-CONF-2010-087 2010/08/19 TOPQ
Calibration of b-tagging using dileptonic top pair events in a combinatorial likelihood approach with the ATLAS experiment ATLAS-CONF-2014-004 2014/02/25 FTAG / TOPQ
Combination of ATLAS and CMS results on the mass of the top quark using up to 4.9 fb-1 of data ATLAS-CONF-2012-095 2012/07/02 TOPQ
Combination of ATLAS and CMS results on the mass of the top-quark using up to 4.9 fb−1 of √s=7 TeV LHC data ATLAS-CONF-2013-102 2013/09/16 TOPQ
Combination of ATLAS and CMS top quark pair cross section measurements in the eμ final state using proton-proton collisions at √s = 8 TeV ATLAS-CONF-2014-054 2014/09/29 TOPQ
Combination of ATLAS and CMS top-quark pair cross section measurements using up to 1.1 fb-1 of data at 7 TeV ATLAS-CONF-2012-134 2012/09/17 TOPQ
Combination of ATLAS and CMS ttbar charge asymmetry measurements using LHC proton-proton collisions at √s = 7 TeV ATLAS-CONF-2014-012 2014/03/24 TOPQ
Combination of cross-section measurements for associated production of a single top-quark and a W boson at √s = 8 TeV with the ATLAS and CMS experiments ATLAS-CONF-2016-023 2016/05/16 TOPQ
Combination of cross-section measurements for associated production of a single top-quark and a W boson at √s =8 TeV with the ATLAS and CMS experiments ATLAS-CONF-2014-052 2014/09/29 TOPQ
Combination of single top-quark cross-section measurements in the t-channel at √s = 8 TeV with the ATLAS and CMS experiments ATLAS-CONF-2013-098 2013/09/15 TOPQ
Combination of the ATLAS and CMS measurements of the W-boson polarization in top-quark decays ATLAS-CONF-2013-033 2013/03/12 TOPQ
Constraints on Anomalous μ+μ+ and Like-sign Top Quark Pair Production ATLAS-CONF-2011-139 2011/09/26 EXOT / TOPQ
Determination of the Top Quark Mass with a Template Method in the All-Hadronic Decay Channel using 2.04 fb-1 of ATLAS Data ATLAS-CONF-2012-030 2012/03/14 TOPQ
Determination of the Top-Quark Mass from the ttbar Cross Section Measurement in pp Collisions at √s = 7 TeV with the ATLAS detector ATLAS-CONF-2011-054 2011/03/31 TOPQ
Determination of the top-quark pole mass using tt̄ +1-jet events collected with the ATLAS experiment in 7 TeV pp collisions ATLAS-CONF-2014-053 2014/09/29 TOPQ
Differential top-antitop production cross-section measurements in pp collisions at √s = 7 TeV as a function of pseudo-top-quark observables in the single-lepton channel using the ATLAS detector. ATLAS-CONF-2014-059 2014/09/28 TOPQ
Direct top-quark decay width measurement in the $tt$ lepton+jets channel at $√s=8$ TeV with the ATLAS experiment ATLAS-CONF-2017-056 2017/07/05 TOPQ
Estimation of non-prompt and fake lepton backgrounds in final states with top quarks produced in proton-proton collisions at √s = 8 TeV with the ATLAS detector ATLAS-CONF-2014-058 2014/09/30 TOPQ
Evidence for single top-quark production in the s-channel in proton-proton collisions at √s=8TeV with the ATLAS detector using the Matrix Element Method ATLAS-CONF-2015-047 2015/09/12 TOPQ
Evidence for the associated production of a vector boson (W, Z) and top quark pair in the dilepton and trilepton channels in pp collision data at √s = 8 TeV collected by the ATLAS detector at the LHC ATLAS-CONF-2014-038 2014/07/04 TOPQ
First combination of Tevatron and LHC measurements of the top-quark mass ATLAS-CONF-2014-008 2014/03/17 TOPQ
Measurement of colour flow using jet-pull observables in $tt$ events with the ATLAS experiment at $√s$ = 13 TeV ATLAS-CONF-2017-069 2017/09/16 TOPQ
Measurement of jets produced in top quark events using the di-lepton final state with 2 $b$-tagged jets in p p collisions at $√s$ = 13 TeV with the ATLAS detector ATLAS-CONF-2015-065 2015/12/15 TOPQ
Measurement of spin correlation in top-antitop quark events and search for top squark pair production in proton-proton collisions at √s = 8 TeV using the ATLAS detector ATLAS-CONF-2014-056 2014/09/26 TOPQ / SUSY
Measurement of spin correlation in top-quark pair production from pp collisions at √s=7 TeV using the ATLAS detector ATLAS-CONF-2011-117 2011/08/17 TOPQ
Measurement of t-Channel Single Top-Quark Production in pp Collisions at √s = 8 TeV with the ATLAS detector ATLAS-CONF-2014-007 2014/03/16 TOPQ
Measurement of t-Channel Single Top-Quark Production in pp Collisions at √s = 8 TeV with the ATLAS detector ATLAS-CONF-2012-132 2012/09/16 TOPQ
Measurement of the charge asymmetry in dileptonic decay of top quark pairs in pp collisions at √s = 7 TeV using the ATLAS detector ATLAS-CONF-2012-057 2012/06/02 TOPQ
Measurement of the charge asymmetry in highly boosted top-quark pair production in √s= 8 TeV pp collision data collected by the ATLAS experiment ATLAS-CONF-2015-048 2015/09/14 TOPQ
Measurement of the charge asymmetry in top quark pair production in pp collisions at √s = 7 TeV using the ATLAS detector ATLAS-CONF-2011-106 2011/08/04 TOPQ
Measurement of the cross section for tt̄+jets production using a kinematic fit method with the ATLAS detector ATLAS-CONF-2012-083 2012/07/02 TOPQ
Measurement of the cross-section for associated production of a top quark and a W boson at √s = 8 TeV with the ATLAS detector ATLAS-CONF-2013-100 2013/09/13 TOPQ
Measurement of the cross-section of the production of a W boson in association with a single top quark with ATLAS at √s = 13TeV ATLAS-CONF-2016-065 2016/08/03 TOPQ
Measurement of the differential cross-section of highly boosted top quarks as a function of their transverse momentum using the ATLAS detector in √s = 8 TeV proton-proton collisions ATLAS-CONF-2014-057 2014/09/26 TOPQ
Measurement of the inclusive $tt$ cross-section in the lepton+jets channel with pp collisions at $√s$ = 8 TeV ATLAS-CONF-2017-054 2017/07/06 TOPQ
Measurement of the inclusive cross-section of single top-quark t-channel production in pp collisions at √s = 13 TeV ATLAS-CONF-2015-079 2015/12/15 TOPQ
Measurement of the inclusive tt̄γ cross section at √s = 7 TeV with the ATLAS detector ATLAS-CONF-2011-153 2011/11/13 TOPQ
Measurement of the jet multiplicity in topâ€“anti-top final states produced in 7 TeV protonâ€“proton collisions with the ATLAS detector ATLAS-CONF-2012-155 2012/11/11 TOPQ
Measurement of the production cross-section of a single top quark in association with a Z boson in proton-proton collisions at 13 TeV with the ATLAS detector ATLAS-CONF-2017-052 2017/06/06 TOPQ
Measurement of the t-channel single top-quark and top-antiquark cross-sections and their ratio in pp collisions at √s = 7 TeV ATLAS-CONF-2012-056 2012/06/02 TOPQ
Measurement of the t-channel Single Top-Quark Production Cross Section in 0.70 fb-1 of pp Collisions at √s = 7 TeV with the ATLAS detector ATLAS-CONF-2011-101 2011/07/20 TOPQ
Measurement of the top quark charge in pp collisions at √s = 7 TeV in the ATLAS experiment ATLAS-CONF-2011-141 2011/09/26 TOPQ
Measurement of the Top Quark Mass from √s = 7 TeV ATLAS Data using a 3-dimensional Template Fit ATLAS-CONF-2013-046 2013/05/15 TOPQ
Measurement of the Top Quark Mass in Dileptonic Top Quark Pair Decays with √s=7 TeV ATLAS Data ATLAS-CONF-2013-077 2013/07/16 TOPQ
Measurement of the top quark mass in the $t \overline{t} →$ lepton+jets channel from $√s=$ 8 TeV ATLAS data ATLAS-CONF-2017-071 2017/09/18 TOPQ
Measurement of the top quark mass in topologies enhanced with single top quarks produced in the t-channel at √s = 8 TeV using the ATLAS experiment ATLAS-CONF-2014-055 2014/09/26 TOPQ
Measurement of the top quark pair cross-section with ATLAS in pp collisions at √s = 7 TeV in the single-lepton channel using b-tagging ATLAS-CONF-2011-035 2011/03/20 TOPQ
Measurement of the top quark pair production charge asymmetry in proton-proton collisions at √s =7 TeV using the ATLAS detector ATLAS-CONF-2013-078 2013/07/18 TOPQ
Measurement of the top quark pair production cross section in pp collisions at √s = 7 TeV in μ+τ final states with ATLAS ATLAS-CONF-2011-119 2011/08/18 TOPQ
Measurement of the top quark pair production cross section in the single-lepton channel with ATLAS in proton-proton collisions at 8 TeV using kinematic fits with b-tagging ATLAS-CONF-2012-149 2012/11/10 TOPQ
Measurement of the top quark pair production cross section with ATLAS in pp collisions at √s=7 TeV in dilepton final states ATLAS-CONF-2011-034 2011/03/21 TOPQ
Measurement of the top quark pair production cross section with ATLAS in pp collisions at √s = 7 TeV in the single-lepton channel using semileptonic b decays ATLAS-CONF-2012-131 2012/09/16 TOPQ
Measurement of the top quark pair production cross-section based on a statistical combination of measurements of dilepton and single-lepton final states at √s = 7 TeV with the ATLAS detector ATLAS-CONF-2011-108 2011/07/20 TOPQ
Measurement of the Top-Quark Mass from 2011 ATLAS Data using the Template Method ATLAS-CONF-2011-120 2011/08/20 TOPQ
Measurement of the Top-Quark Mass using the Template Method in pp Collisions at √s = 7 TeV with the ATLAS detector ATLAS-CONF-2011-033 2011/03/18 TOPQ
Measurement of the top-quark pair production cross-section in pp collisions at √s = 7 TeV in dilepton final states with ATLAS ATLAS-CONF-2011-100 2011/07/20 TOPQ
Measurement of the ttˉ production cross-section in pp collisions at √ s =8 TeV using eμ events with b-tagged jets ATLAS-CONF-2013-097 2013/09/13 TOPQ
Measurement of the tt̄ production cross section in the final state with a hadronically decaying tau lepton and jets using the ATLAS detector ATLAS-CONF-2012-032 2012/03/12 TOPQ
Measurement of the tt̄ production cross-section in pp collisions at √s = 7 TeV using kinematic information of lepton+jets events ATLAS-CONF-2011-121 2011/08/20 TOPQ
Measurement of the tt̄W and tt̄Z cross sections in proton-proton collisions at √s = 13 TeV with the ATLAS detector ATLAS-CONF-2018-047 2018/09/20 TOPQ
Measurement of the tt̄Z and tt̄W production cross sections in multilepton final states using 3.2 fb-1 of pp collisions at 13 TeV at the LHC ATLAS-CONF-2016-003 2016/03/14 TOPQ
Measurement of the ttbar production cross-section in pp collisions at √s=13 TeV using emu events with b-tagged jets ATLAS-CONF-2015-033 2015/07/25 TOPQ
Measurement of the W boson polarisation in top quark decays in 0.70 fb-1 of pp collisions at √s = 7 TeV with the ATLAS detector ATLAS-CONF-2011-122 2011/08/21 TOPQ
Measurement of the W-boson polarisation in top quark decays in pp collision data at √s = 7 TeV using the ATLAS detector ATLAS-CONF-2011-037 2011/03/22 TOPQ
Measurement of top quark pair differential cross-sections in the l+jets channel in pp collisions at √s=7 TeV using the ATLAS detector ATLAS-CONF-2013-099 2013/09/15 TOPQ
Measurement of top quark polarisation in tt̄ events with the ATLAS detector in proton-proton collisions at √s = 7 TeV ATLAS-CONF-2012-133 2012/09/17 TOPQ
Measurement of tt̄ production in the all-hadronic channel in 1.02 fb-1 of pp collisions at √s = 7 TeV with the ATLAS detector ATLAS-CONF-2011-140 2011/09/24 TOPQ
Measurement of tt̄ production in the all-hadronic channel in 4.7 fb-1 of pp collisions at √s = 7 TeV with the ATLAS detector ATLAS-CONF-2012-031 2012/03/14 TOPQ
Measurements of fiducial and differential cross-sections of $tt$ production with additional heavy-flavour jets in proton-proton collisions at √s = 13 TeV with the ATLAS detector ATLAS-CONF-2018-029 2018/07/02 TOPQ
Measurements of inclusive and differential cross-sections of $ttγ$ production in leptonic final states in a fiducial volume at $√s=13~\text{TeV}$ in ATLAS ATLAS-CONF-2018-048 2018/09/20 TOPQ
Measurements of spin correlation in top-antitop quark events from proton-proton collisions at √s = 7 TeV using the ATLAS detector ATLAS-CONF-2013-101 2013/09/16 TOPQ
Measurements of the $tt$ production cross-section in the dilepton and lepton-plus-jets channels and of the ratio of the $tt$ and Z boson cross-sections in pp collisions at $√s=13$ TeV with the ATLAS detector ATLAS-CONF-2015-049 2015/09/14 TOPQ
Measurements of top quark spin observables in tt̄ events using dilepton final states at √s = 8 TeV with the ATLAS detector ATLAS-CONF-2016-099 2016/09/19 TOPQ
Measurements of top-quark pair differential cross-sections in the lepton+jets channel in pp collisions at √s=13 TeV using the ATLAS detector ATLAS-CONF-2016-040 2016/08/01 TOPQ
Measurements of top-quark pair spin correlations in the eμ channel at √s=13 TeV using pp collisions in the ATLAS detector. ATLAS-CONF-2018-027 2018/07/04 TOPQ
Measuring the b-tag efficiency in a ttbar sample with 4.7 fb^{-1} of data from the ATLAS detector ATLAS-CONF-2012-097 2012/06/06 FTAG / TOPQ
Observation of t-channel Single Top-Quark Production in pp Collisions at √s = 7 TeV with the ATLAS detector ATLAS-CONF-2011-088 2011/06/05 TOPQ
Probing the Wtb vertex structure with t-channel single top-quark events in pp collisions at √s = 8 TeV with the ATLAS detector ATLAS-CONF-2016-097 2016/09/16 TOPQ
Reconstructed jet multiplicities from the top-quark pair decays and associated jets in pp collisions at √s = 7 TeV measured with the ATLAS detector at the LHC ATLAS-CONF-2011-142 2011/09/26 TOPQ
Reconstruction of $J/\psi$ mesons in $t{\bar t}$ final states in proton-proton collisions at $√s=8$ TeV with the ATLAS detector ATLAS-CONF-2015-040 2015/08/19 TOPQ
Search for Anomalous Missing ET in tt Events ATLAS-CONF-2011-036 2011/03/18 TOPQ
Search for b' and 4-top production in the same-sign dilepton channel using 8 TeV data ATLAS-CONF-2013-051 2013/05/13 EXOT / TOPQ
Search for charged lepton-flavour violation in top-quark decays at the LHC with the ATLAS detector ATLAS-CONF-2018-044 2018/09/13 TOPQ
Search for CP violation in single top quark events in pp collisions at √s = 7 TeV with the ATLAS detector ATLAS-CONF-2013-032 2013/03/11 TOPQ
Search for exotic same-sign dilepton signatures (b' quark, T5/3 and four top quarks production) in 4.7 fb-1 of pp collisions at √s=7 TeV with the ATLAS detector ATLAS-CONF-2012-130 2012/09/13 EXOT / TOPQ
Search for FCNC Top Quark Processes at √s = 7 TeV with the ATLAS Detector ATLAS-CONF-2011-061 2011/04/12 TOPQ
Search for flavour-changing neutral current top quark decays t$→$qZ in proton-proton collisions at $√s$=13 TeV with the ATLAS Detector ATLAS-CONF-2017-070/ 2017/09/18 TOPQ
Search for four-top-quark production in final states with one charged lepton and multiple jets using 3.2 fb$^{-1}$ of proton-proton collisions at $√s=13$ TeV with the ATLAS detector at the LHC ATLAS-CONF-2016-020 2016/04/07 EXOT / TOPQ
Search for pair production of new heavy quarks that decay to a Z boson and a third generation quark in pp collisions at √s=8 TeV with the ATLAS detector ATLAS-CONF-2014-036 2014/07/02 EXOT / TOPQ
Search for pair production of new heavy quarks that decay to a Z boson and a third generation quark in pp collisions at √s=8 TeV with the ATLAS detector ATLAS-CONF-2013-056 2013/06/20 EXOT / TOPQ
Search for resonances decaying into top quark pairs using fully hadronic decays in pp collisions with ATLAS at √s = 7 TeV ATLAS-CONF-2012-102 2012/07/24 TOPQ / EXOT
Search for s-channel Single Top-Quark Production in pp Collisions at √s = 7 TeV ATLAS-CONF-2011-118 2011/08/17 TOPQ
Search for single top-quark production via FCNC in strong interactions in √s = 8 TeV ATLAS data ATLAS-CONF-2013-063 2013/07/09 TOPQ
Search for top pair candidate events in ATLAS at √s=7 TeV ATLAS-CONF-2010-063 2010/07/21 TOPQ
Search for top quark decays $t → Hq$ with 36 fb$^{-1}$ of $pp$ collision data at $√s=13$ TeV with the ATLAS detector ATLAS-CONF-2018-049 2018/09/20 TOPQ
Search for top-jet resonances in the lepton+jets channel of tt̄ + jets events with the ATLAS detector in 4.7 fb-1 of pp collisions at √s = 7 TeV ATLAS-CONF-2012-096 2012/07/04 TOPQ / EXOT
Search for ttbar production in the all-hadronic channel in ATLAS with √s = 7 TeV data ATLAS-CONF-2011-066 2011/04/24 TOPQ
Search for Vector Like Quarks in the lepton+jets final state using Ht/b+X and Wb+X signatures ATLAS-CONF-2015-012 2015/03/17 EXOT / TOPQ
Search for VLQ decay to WbWb in the lepton+jet channel using 8 TeV data ATLAS-CONF-2013-060 2013/06/21 EXOT / TOPQ
Search for VLQ decay to Ht+X in the lepton+jet channel using 8 TeV data ATLAS-CONF-2013-018 2013/03/07 EXOT / TOPQ
Search for W' → tb̄ in proton-proton collisions at a centre-of-mass energy of √s = 8 TeV with the ATLAS detector ATLAS-CONF-2013-050 2013/05/13 TOPQ / EXOT
Search for Wt associated production in dilepton final states with 0.70 fb-1 of 7 TeV pp collision data in ATLAS ATLAS-CONF-2011-104 2011/07/20 TOPQ
Searches for Single Top-Quark Production with the ATLAS Detector in pp Collisions at √s = 7 TeV ATLAS-CONF-2011-027 2011/03/13 TOPQ
Statistical combination of top quark pair production cross-section measurements using dilepton, single-lepton, and all-hadronic final states at √s = 7 TeV with the ATLAS detector ATLAS-CONF-2012-024 2012/03/05 TOPQ
Top Quark Pair Production Cross-section Measurement in ATLAS in the Single Lepton+Jets Channel without b-tagging ATLAS-CONF-2011-023 2011/03/13 TOPQ
Top-quark mass measurement in the all-hadronic $tt$ decay channel at $√s=8$ TeV with the ATLAS detector ATLAS-CONF-2016-064 2016/08/08 TOPQ
Top-quark mass measurement in the eμ channel using the mT2 variable at ATLAS ATLAS-CONF-2012-082 2012/06/30 TOPQ
Number of ConfNotes: 121

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#### Combust

A planet is termed combust (burned) when it is in conjunction with the Sun and therefore hidden from sight by the light of the Sun. Traditionally this is a serious affliction and implies that the planet is weakened or restricted in power. It is often used to signify someone in fear or feeling overwhelmed, or to represent things that are kept hidden from sight.

William Lilly stated that the combust planet should be within 8° 30' – beyond this distance, but within 17° of the Sun, the planet is said to be under the Sun’s beams. This condition is debilitating, but not as severe as combustion.

More ancient authors generally used 15° to note the distance of being under the Sun's beams, and were less specific about the limits of combustion - generally, the expectation was that the closer the planets is to the Sun, the more intense and debilitating the effect is expected to be.


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# What are the current obstacles to experimentally testing quantum pseudo telepathy?

Quantum pseudo-telepathy refers to how, in some specific coordination games, isolated players can do better when they have pre-shared some entangled qubits. I understand how it works in theory and wrote about it, but I'm curious if it's something we can do in practice.

How close are we to being able to do real life runs of these games, where a quantum strategy is used to win more often than possible with classical strategies? I know we don't have quantum computers with more than a few qubits now, but the game in the wikipedia article seems to only require two per player.

What if we relax some of the constraints, like just saying "you've been isolated" instead of actually separating the players/machines by large distances? Or allowing the entangled qubits to be generated on demand instead of ahead of time?

If we can't do it yet, or in the near future, what are the obstacles?

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What's the difference between "telepathy" and good ol' violation of Bell inequalities? – Emilio Pisanty Sep 28 '13 at 16:42
@Emilio I mostly just find it to be an interesting application. More relatable than measurements agreeing more often than you'd expect, because it directly shows you can use it. In this review paper Gilles Brassard et al. mention it's a good way to demonstrate QM is correct. – Craig Gidney Sep 28 '13 at 17:09
I'm not saying I don't like it. But it does feel like it's a re-packaging of (possibly a specific class of) what's normally known as Bell inequality violations. – Emilio Pisanty Sep 28 '13 at 17:22

So far, scientists have been able to test spatially separated entanglement between 2 parties, and recently even between 3 parties. The latter experiment (link)(without paywall) states the following challenges:

No attempts have been made to close locality loopholes in Bell experiments involving three or more particles. The primary reason for this is source brightness. While entangled photon pairs have been generated and detected at rates in excess of 1 MHz, entangled photon triplets have only been observed at rates on the order of, necessitating long measurement times. In addition, further experimental challenges include high sensitivity to loss, causality relations requiring a complex experimental set-up and demanding stability requirements.

However, these type of experiments are interested in measuring the violation of classical bounds (e.g., proving that no classical hidden variable theories are possible, thus confirming Quantum Mechanics), such as Bell's or Mermin's inequalities. Your question is why games have not yet been experimentally demonstrated, but it turns out that testing inequalities is exactly the same.

Edit: I realized only later that quantum games and Bell-type inequalities are actually the exact same. Let me try to explain the equivalence below, and then give an example of what an experiment would look like in the "game-language".

A bell-type inequality says something about expectation values of measurements in pre-defined bases. The equivalent game dictates that players receive a random question, and depending on the question, they measure in a particular basis.

Now, you will find that the number of "expectation values" and the "number of questions" are the exact same. Moreover, the corresponding measurements turn out to be along the same basis (assuming we start from the same shared state)! Thus, when performing an optimal stategy multiple times, the players of a pseudo-telepathy game basically sample the expectation values.

The tricky part is that the expectation values in Bell-inequalities do not correspond to winning probablities $\omega$ (a number between 0 and 1), but to so-called "biasses" $\varepsilon$, which is simply $\omega$ rescaled to lie between (-1,1) (indeed, the values of a quantum measurement outcome!).

$$\varepsilon = 2 \omega - 1$$

Take for example the CHSH game, where the optimal winning probabilitiy $\omega$ equals $\cos^2(\pi/8) = 1/2 + 1/\sqrt{8}$ (on average, per question). Turning these into biasses, and adding all 4 questions together, gives the well-known bound $2\sqrt{2}$, which is indeed the maximal violation of the Bell-inequality.

Now, you can simply look at any Bell-inequality experiment, turn it into the "game-language", and see how well the "players" scored, on average. For example, in the 3-qubit Mermin inequalities paper, take a look at figure 2. It shows the "biasses" per question (a,b,c). The "worst" expectation found was −0.655 where they should've measured -1, which means a bias of +0.655. This corresponds to a "winning probability" of 0.8275 on that question. Considering that this was the question on which they performed the worst, they clearly did better than the 0.75 classical bound.

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The free version of the paper is here : arxiv.org/abs/1309.1379 – Frédéric Grosshans Mar 21 at 15:24


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Definitions

# Group (mathematics)

In mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and operation must satisfy a few conditions called group axioms, namely associativity, identity and inverse elements. While these are familiar from many mathematical structures, such as number systems—for example, the integers endowed with the addition operation form a group—the formulation of the axioms is detached from the concrete nature of the group and its operation. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond. The ubiquity of groups in numerous areas—both within and outside mathematics—makes them a central organizing principle of contemporary mathematics.

Groups share a fundamental kinship with the notion of symmetry. A symmetry group encodes symmetry features of a geometrical object: it consists of the set of transformations that leave the object unchanged, and the operation of combining two such transformations by performing one after the other. Such symmetry groups, particularly the continuous Lie groups, play an important role in many academic disciplines. Matrix groups, for example, can be used to understand fundamental physical laws underlying special relativity and symmetry phenomena in molecular chemistry.

The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—a very active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups completed in 1983.

## Definition and illustration

### First example: the integers

One of the most familiar groups is the set of integers Z containing the numbers
..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...
The following properties of integer addition serve as a model for the abstract group axioms given in the definition below.

1. For any two integers a and b, the sum a + b is also an integer. In other words, the process of adding integers two at a time can never yield a result that is not an integer. This property is known as closure under addition.
2. For all integers a, b and c, (a + b) + c = a + (b + c). Expressed in words, adding a to b first, and then adding c to their sum gives the same final result as adding the sum of b and c to a, a property known as associativity.
3. If a is any integer, then 0 + a = a + 0 = a. Zero is called the identity element of addition because adding it to any integer returns the same integer.
4. For every integer a, there is an integer b such that a + b = b + a = 0. The integer b is called the inverse element of the integer a and is denoted −a.

### Definition

The integers, together with the operation "+", form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures without dealing with every concrete case separately, the following abstract definition is developed to encompass the above example along with many others, one of which is the symmetry group detailed below. A group is a set, G, together with an operation "•" that combines any two elements a and b to form another element denoted . The symbol "•" is a general placeholder for a concretely given operation, such as the addition above. To qualify as a group, the set and operation, , must satisfy four requirements known as the group axioms:

 1 Closure. For all a, b in G, the result of the operation a • b is also in G. 2 Associativity. For all a, b and c in G, the equation (a • b) • c = a • (b • c) holds. 3 Identity element. There exists an element e in G, such that for all elements a in G, the equation e • a = a • e = a holds. 4 Inverse element. For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element.

The order in which the group operation is carried out can be significant. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation

ab = ba
may not always be true. This equation does always hold in the group of integers under addition, because a + b = b + a for any two integers (commutativity of addition). However, it does not always hold in the symmetry group below. Groups for which the equation ab = ba always holds are called abelian (in honor of Niels Abel). Thus, the integer addition group is abelian, but the following symmetry group is not.

### Second example: a symmetry group

The symmetries (i.e. rotations and reflections) of a square form a group called a dihedral group, and denoted D4. The following symmetries occur:
 id (keeping it as is) r1 (rotation by 90° right) r2 (rotation by 180° right) r3 (rotation by 270° right) fv (vertical flip) fh (horizontal flip) fd (diagonal flip) fc (counter-diagonal flip) The elements of the symmetry group of the square (D4). The vertices are colored and numbered only to visualize the operations.
* the identity operation leaving everything unchanged, denoted id;
* rotations of the square by 90° right, 180° right, and 270° right, denoted by r1, r2 and r3, respectively;
* reflections about the vertical and horizontal middle line (fh and fv), or through the two diagonals (fd and fc).

Any two symmetries a and b can be composed, i.e. applied one after another. The result of performing first a and then b is written symbolically from right to left as

ba ("apply the symmetry b after performing the symmetry a". The right-to-left notation stems from composition of functions).
The group table on the right lists the results of all such compositions possible. For example, rotating by 270° right (r3) and then flipping horizontally (fh) is the same as performing a reflection along the diagonal (fd). Using the above symbols, highlighted in blue in the group table:
fh • r3 = fd.

Group table of D4
id r1 r2 r3 fv fh fd fc
id id r1 r2 r3 fv fh fd fc
r1 r1 r2 r3 id fc fd fv fh
r2 r2 r3 id r1 fh fv fc fd
r3 r3 id r1 r2 fd fc

fh fv
fv fv fd fh fc id r2 r1 r3
fh fh fc fv fd r2 id r3 r1
fd fd fh fc fv r3 r1 id r2
fc fc fv fd fh r1 r3 r2 id
The elements id, r1, r2, and r3 form a subgroup, highlighted in red (upper left region). A left and right coset of this subgroup is highlighted in green (in the last row) and yellow (last column), respectively.
Given this set of symmetries and the described operation, the group axioms can be understood as follows:
1. The closure axiom demands that the composition ba of any two symmetries a and b is still a symmetry. Another example for the group operation is
r3 • fh = fc,
i.e. rotating 270° right after flipping horizontally equals flipping along the counter-diagonal (fc). Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table.

2. The associativity constraint deals with composing more than two symmetries: given three elements a, b and c of D4, there are two possible ways of computing "a then b then c". The requirement
(ab) • c = a • (bc)
means that the composition of the three elements is independent of the priority of the operations, i.e. composing first a after b, and c to the result thereof amounts to performing a after the composition of b and c. For example, (fd • fv) • r2 = fd • (fv • r2) can be checked using the group table at the right
 (fd • fv) • r2 = r3 • r2 = r1, which equals fd • (fv • r2) = fd • fh = r1.

3. The identity element is the symmetry id leaving everything unchanged: for any symmetry a, performing id after a (or a after id) equals a, in symbolic form,
id • a = a,
a • id = a.
4. An inverse element undoes the transformation of some other element. Every symmetry can be undone: each of transformations—identity id, the flips fh, fv, fd, fc and the 180° rotation r2—is its own inverse, because performing each one twice brings the square back to its original orientation. The rotations r3 and r1 are each other's inverse, because rotating one way and then by the same angle the other way leaves the square unchanged. In symbols,
fh • fh = id,
r3 • r1 = r1 • r3 = id.

In contrast to the group of integers above, where the order of the operation is irrelevant, it does matter in D4: fh • r1 = fc but r1 • fh = fd. In other words, D4 is not abelian, which makes the group structure more difficult than the integers introduced first.

## History

The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 (1854) gives the first abstract definition of a finite group.

Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.

The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker. In 1847, Ernst Kummer led early attempts to prove Fermat's Last Theorem to a climax by developing groups describing factorization into prime numbers.

The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) gave the first statement of the modern definition of an abstract group. As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was pushed by Hermann Weyl, Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by pivotal work of Armand Borel and Jacques Tits.

The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, classified all finite simple groups in 1982. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification. These days, group theory is still a highly active mathematical branch crucially impacting many other fields.

## Simple consequences of the group axioms

Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory. For example, repeated applications of the associativity axiom show that the unambiguity of
abc = (ab) • c = a • (bc)
generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted.

The axioms may be weakened to assert only the existence of a left identity and left inverses. Both can be shown to be actually two-sided, so the resulting definition is equivalent to the one given above.

### Uniqueness of identity element and inverses

Two important consequences of the group axioms are the uniqueness of the identity element and the uniqueness of inverse elements. There can be only one identity element in a group, and each element in a group has exactly one inverse element. Thus, it is customary to speak of the identity, and the inverse of an element.

To prove the uniqueness of an inverse element of a, suppose that a has two inverses, denoted l and r. Then

 l = l • e as e is the identity element = l • (a • r) because r is an inverse of a, so e = a • r = (l • a) • r by associativity, which allows to rearrange the parentheses = e • r since l is an inverse of a, i.e. l • a = e = r for e is the identity element

Hence the two extremal terms l and r are connected by a chain of equalities, so they agree. In other words there is only one inverse element of a.

### Division

In groups, it is possible to perform division: given elements a and b of the group G, there is exactly one solution x in G to the equation xa = b. In fact, right multiplication of the equation by a−1 gives the solution x = xaa−1 = ba−1. Similarly there is exactly one solution y in G to the equation ay = b, namely y = a−1b. In general, x and y need not agree.

## Basic concepts

To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed. There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which for example sets—being "structureless"—don't have) constructions related to groups have to be compatible with the group operation. This compatibility manifests itself in the following notions in various ways. For example, groups can be related to each other via functions called group homomorphisms. By the mentioned principle, they are required to respect the group structures in a precise sense. The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups. The principle of "preserving structures"—a recurring topic in mathematics throughout—is an instance of working in a category, in this case the category of groups.

### Group homomorphisms

Group homomorphisms are functions that preserve group structure. A function a: GH between two groups is a homomorphism if the equation
a(gk) = a(g) • a(k).
holds for all elements g, k in G, i.e. the result is the same when performing the group operation after or before applying the map a. This requirement ensures that a(1G) = 1H, and also a(g)−1 = a(g−1) for all g in G. Thus a group homomorphism respects all the structure of G provided by the group axioms.

Two groups G and H are called isomorphic if there exist group homomorphisms a: GH and b: HG, such that applying the two functions one after another (in each of the two possible orders) equal the identity function of G and H, respectively. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of view, isomorphic groups carry the same information. For example, proving that gg = 1 for some element g of G is equivalent to proving that a(g) • a(g) = 1, because applying a to the first equality yields the second, and applying b to the second gives back the first.

### Subgroups

Informally, a subgroup is a group H contained within a bigger one, G. Concretely, the identity element of G is contained in H, and whenever h1 and h2 are in H, then so are and h1−1, so the elements of H, equipped with the group operation on G restricted to H, form indeed a group.

In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the rotation in the opposite direction. The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that for all elements g, hH. Knowing the subgroups is important in understanding the group as a whole.

Given any subset S of a group G, the subgroup generated by S consists of products of elements of S and their inverses. It is the smallest subgroup of G containing S. In the introductory example above, the subgroup generated by r2 and fv consists of these two elements, the identity element id and fh = fv • r2. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.

### Cosets

In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a flip is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further flips), i.e. the rotation operations are irrelevant to the question whether a flip has been performed. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In symbolic terms, the left and right coset of H containing g are

gH = {gh, hH} and Hg = {hg, hH}, respectively.

The cosets of any subgroup H form a partition of G; that is, two left cosets are either equal or have an empty intersection. The first case g1H = g2H happens precisely when , i.e. if the two elements differ by an element of H. Similar considerations apply to the right cosets of H. The left and right cosets of H may or may not be equal. If they are, i.e. for all g in G, gH = Hg, then H is said to be a normal subgroup. One may then simply refer to N as the set of cosets.

In D4, the introductory symmetry group, the left cosets gR of the subgroup R consisting of the rotations are either equal to R, if g is an element of R itself, or otherwise equal to U = fvR = {fv, fd, fh, fc} (highlighted in green). The subgroup R is also normal, because fvR = U = Rfv and similarly for any element other than fv.

### Quotient groups

In addition to disregarding the internal structure of a subgroup by considering its cosets, it is desirable to endow this coarser entity with a group law called quotient group or factor group. For this to be possible, the subgroup has to be normal. Given any normal subgroup N, the quotient group is defined by
G / N = {gN, gG}, "G modulo N
This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map that associates to any element g its coset gN be a group homomorphism, or by general abstract considerations called universal properties. The coset eN = N serves as the identity in this group, and the inverse of Ng in the quotient group is (gN)−1 = (g−1)N.

R U
R R U
U U R
Group table of the quotient group .
The elements of the quotient group are R itself, which represents the identity, and U = fvR. The group operation on the quotient is shown at the right. For example, UU = fvR • fvR = (fv • fv)R = R. Both the subgroup R = {id, r1, r2, r3}, as well as the corresponding quotient are abelian, whereas D4 is not abelian. Building bigger groups by smaller ones, such as D4 from its subgroup R and the quotient is abstracted by a notion called semidirect product.

Quotient and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) flip), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

r 4 = f 2 = (rf)2 = 1,
the group is completely described. A presentation of a group can also be used to construct the Cayley graph, a device used to graphically capture discrete groups.

Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map , i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map . Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.

## Examples and applications

Examples and applications of groups abound. A starting point is the group Z of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra.

Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups. For example, the image at the right depicts elements of the fundamental group of a plane minus a point. Elements of the group are given by loops in this area. The blue loop is considered null-homotopic (and thus irrelevant), because it can be shrunk to a point. The presence of the hole prevents the orange loop from being shrunk. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.

In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups. Further branches crucially applying groups include algebraic geometry and number theory.

In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory, in particular when implemented for finite groups. Applications of group theory are not restricted to mathematics; sciences such as physics, chemistry and computer science benefit from the concept.

### Numbers

Many number systems, such as the integers and the rationals enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields.

#### Integers

The group of integers Z under addition, denoted (Z, +), has been described above. The integers, with the operation of multiplication instead of addition, (Z, ·) do not form a group. The closure, associativity and identity axioms are satisfied, but inverses do not exist: for example, a = 2 is an integer, but the only solution to the equation a · b = 1 in this case is b = 1/2, which is a rational number, but not an integer. Hence not every element of Z has a (multiplicative) inverse.

#### Rationals

The desire for the existence of multiplicative inverses suggests considering fractions
$frac\left\{a\right\}\left\{b\right\}$.
Fractions of integers (with b nonzero) are known as rational numbers. The set of all such fractions is commonly denoted Q. There is still a minor obstacle for the rationals with multiplication, being a group: because the rational number 0 does not have a multiplicative inverse (i.e., there is no x such that x · 0 = 1), (Q, ·) is still not a group.

However, the set of all nonzero rational numbers Q {0} = {qQ, q ≠ 0} does form an abelian group under multiplication, denoted {{nowrap|(Q {0}, ·)}}. Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of a/b is b/a, therefore the axiom of the inverse element is satisfied.

The rational numbers (including 0) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and—if division is possible, such as in Q—fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.

#### Nonzero integers modulo a prime

For any prime number p, modular arithmetic furnishes the multiplicative group of integers modulo p. Its elements are integers not divisible by p, considered modulo p, i.e. two numbers are considered equivalent if their difference is divisible by p. For example, if p = 5, there are exactly four group elements 1, 2, 3, 4: multiples of 5 are excluded and 6 and −4 are both equivalent to 1 etc. The group operation is given by multiplication. Therefore, 4 · 4 = 1, because the usual product 16 is equivalent to 1, for 5 divides 16 − 1 = 15, denoted
16 ≡ 1 (mod 5).
The primality of p ensures that the product of two integers neither of which is divisible by p is not divisible by p either, hence the indicated set of classes is closed under multiplication. The identity element is 1, as usual for a multiplicative group, and the associativity follows from the corresponding property of integers. Finally, the inverse element axiom requires that given an integer a not divisible by p, there exists an integer b such that
a · b ≡ 1 (mod p), i.e. p divides the difference .
The inverse b can be found by using Bézout's identity and the fact that the greatest common divisor equals 1. In the case p = 5 above, the inverse of 4 is 4, and the inverse of 3 is 2, as 3 · 2 = 6 ≡ 1 (mod 5). Hence all group axioms are fulfilled. Actually, this example is similar to (Q{0}, ·) above, because it turns out to be the multiplicative group of nonzero elements in the finite field Fp, denoted Fp×. These groups are crucial to public-key cryptography.

### Cyclic groups

A cyclic group is a group all of whose elements are powers (when the group operation is written additively, the term 'multiple' can be used) of a particular element a. In multiplicative notation, the elements of the group are:

..., a−3, a−2, a−1, a0 = e, a, a2, a3, ...,
where a2 means aa, and a−3 stands for a−1a−1a−1=(aaa)−1 etc. Such an element a is called a generator or a primitive element of the group.

A typical example for this class of groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1 (and whose operation is multiplication). Any cyclic group with n elements is isomorphic to this group. Using some field theory, the group Fp× can be shown to be cyclic: for p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1. An infinite cyclic group is isomorphic to (Z, +), the group of integers under addition introduced above. As these two prototypes are both abelian, so is any cyclic group.

The study of abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian.

### Symmetry groups

Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry. Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.

In chemical fields, such as crystallography, space groups and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties. For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.

Not only are groups useful to assess the implications of symmetries in molecules, but surprisingly they also predict that molecules sometimes can change symmetry. The Jahn-Teller effect is a distortion of a molecule of high symmetry when it adopts a particular ground state of lower symmetry from a set of possible ground states that are related to each other by the symmetry operations of the molecule.

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons.

 Buckminsterfullerene displaysicosahedral symmetry. Ammonia, NH3. Its symmetry group is of order 6, generated by a 120° rotation and a reflection. Cubane C8H8 features octahedral symmetry. complex ion, [Cu(O[[hydrogen>H2)6]2+]]. Compared to a perfectly symmetrical shape, the molecule is vertically dilated by about 22% (Jahn-Teller effect). The (2,3,7) triangle group, a hyperbolic group, acts on this tiling of the hyperbolic plane.

Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved. Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.

### General linear group and representation theory

Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.

Representation theory is both an application of the group concept and important for a deeper understanding of groups. It studies the group by its group actions on other spaces. A broad class of group representations are linear representations, i.e. the group is acting on a vector space, such as the three-dimensional Euclidean space R3. A representation of G on an n-dimensional real vector space is simply a group homomorphism

ρ: GGL(n, R)
from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.

Given a group action, this gives further means to study the object being acted on. On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups.

### Galois groups

Galois groups have been developed to help solving polynomial equations by capturing their symmetry features. For example, the solutions of the quadratic equation ax2 + bx + c = 0 are given by
$x = frac\left\{-b pm sqrt \left\{b^2-4ac\right\}\right\}\left\{2a\right\}.$
Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups (in particular their solvability) associated to polynomials give a criterion which polynomials do have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.

The problem can be dealt with even more neatly using field theory: considering the splitting field of a polynomial shifts the problem into the realm of field theory. Modern Galois theory generalizes the above type of Galois groups to field extensions and establishes—via the fundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.

## Finite groups

A group is called finite if it has a finite number of elements. The number of elements is called the order of the group G. An important class is the symmetric groups SN, the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible swaps of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N (Cayley's theorem). Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle.

The order of an element a in a group G is the least positive integer n such that an = e, where an represents $underbrace\left\{a cdot ldots cdot a\right\}_n$, i.e. application of the operation • to n copies of a. (If • represents multiplication, then an corresponds to the nth power of a.) In infinite groups, such an n may not exist, in which case the order of a is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element.

More sophisticated counting techniques, for example counting cosets, yield more precise statements about finite groups: Lagrange's Theorem states that for a finite group G the order of any finite subgroup H divides the order of G. The Sylow theorems give a partial converse.

The dihedral group (discussed above) is a finite group of order 8. The order of r1 is 4, as is the order of the subgroup R it generates (see above). The order of the reflection elements fv etc. is 2. Both orders divide 8, as predicted by Lagrange's Theorem. The groups Fp× above have order .

### Classification of finite simple groups

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim quickly leads to difficult and profound mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups. An intermediate step is the classification of finite simple groups. A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself. The Jordan-Hölder theorem exhibits simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded to prove the monstrous moonshine conjectures, a surprising and deep relation of the largest finite simple sporadic group—the "monster group"—with certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Many groups are simultaneously groups and examples of other mathematical structures. In the language of category theory, they are group objects in a category, meaning that they are objects (that is, examples of another mathematical structure) which come with transformations (called morphisms) that mimic the group axioms. For example, every group (as defined above) is also a set, so a group is a group object in the category of sets.

### Topological groups

Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the reals R under addition, {{nowrap|(R {0}, ·)}}, and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:

$int f\left(x\right),dx = int f\left(x+c\right),dx$
for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

### Lie groups

Lie groups (in honor of Sophus Lie) are groups which also have a manifold structure, i.e. they are spaces looking locally like some Euclidean space of the appropriate dimension. Again, the additional structure, here the manifold structure, has to be compatible, i.e. the maps corresponding to multiplication and the inverse have to be smooth. A standard example is the general linear group introduced above: it is an open subset of the space of all n-by-n matrices, because it is given by the inequality
det (A) ≠ 0,
where A denotes an n-by-n matrix.

Lie groups are of fundamental importance in physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

## Generalizations

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication {{nowrap|(Z {0}, ·)}}, see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as {{nowrap|(Q {0}, ·)}} is derived from {{nowrap|(Z {0}, ·)}}, known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid. The table gives a list of several structures generalizing groups.

## References

### General references

• , Chapter 5 provides a layman-accessible explanation of groups.
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### Historical references

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12 questions linked to/from FIR filter with linear phase, 4 types
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### Why do linear phase filters have symmetric impulse responses?

It was given as a fact that linear phase filters have symmetric impulse responses, but I don't see why that has to be true. Can somebody please explain or prove this?
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### Are the 4 types of linear phase FIR filters the only linear phase FIR filters one can come up with?

Basically my question is the same as the title: Are the 4 types of linear phase FIR filters the only linear phase FIR filters one can come up with? And if so, then why? I'm pretty sure that they are ...
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### why type 2 FIR cannot be used to design high pass filter

Kindly explain why type 2 FIR filter cannot be used to design high pass filter.
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### Delay in Savitzky-Golay filtering

I am applying a Savitzky-Golay filter to a signal, using the scipy function. I need to calculate the lag of the filtered signal, and how much is it behind the original signal. Could someone shed ...
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### Design one of 4 type of FIR filter by window function?

There're 4 type of FIR to chose from symetry ,even symetry ,odd antisymetry, even antysymetry, odd Also, there are many type of window like rectangular, Hanning, Kaiser,etc So how to use a window ...
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### How to prove a filter has linear phase response?

I have designed an FIR filter to have linear phase response using odd-symmetry design. The coefficients of this filter are {2,1,3,1,0,-1,-3,-1,-2}. I am now being asked to prove it has linear phase ...