Group theory

Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G.The index is denoted or or .Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula (interpret the quantities as cardinal numbers if some of them are infinite).Thus the index measures the "relative sizes" of G and H. For example, let be the group of integers under addition, and let be the subgroup consisting of the even integers. Then has two cosets in , namely the set of even integers and the set of odd integers, so the index is 2. More generally, for any positive integer n. When G is finite, the formula may be written as , and it impliesLagrange's theorem that divides . When G is infinite, is a nonzero cardinal number that may be finite or infinite.For example, , but is infinite. If N is a normal subgroup of G, then is equal to the order of the quotient group , since the underlying set of is the set of cosets of N in G. (Wikipedia).

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Subgroups abstract algebra

In this tutorial we define a subgroup and prove two theorem that help us identify a subgroup. These proofs are simple to understand. There are also two examples of subgroups.

From playlist Abstract algebra

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Definition of a Subgroup in Abstract Algebra with Examples of Subgroups

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Definition of a Subgroup in Abstract Algebra with Examples of Subgroups

From playlist Abstract Algebra

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Abstract Algebra | The notion of a subgroup.

We present the definition of a subgroup and give some examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

From playlist Abstract Algebra

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Abstract Algebra | Cyclic Subgroups

We define the notion of a cyclic subgroup and give a few examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

From playlist Abstract Algebra

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Percentiles, Deciles, Quartiles

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From playlist Unit 1: Descriptive Statistics

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Abstract Algebra | Normal Subgroups

We give the definition of a normal subgroup and give some examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

From playlist Abstract Algebra

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Lagrange's Theorem and Index of Subgroups | Abstract Algebra

We introduce Lagrange's theorem, showing why it is true and follows from previously proven results about cosets. We also investigate groups of prime order, seeing how Lagrange's theorem informs us about every group of prime order - in particular it tells us that any group of prime order p

From playlist Abstract Algebra

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Populations, Samples, Parameters, and Statistics

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Populations, Samples, Parameters, and Statistics

From playlist Statistics

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GT2. Definition of Subgroup

Abstract Algebra: We define the notion of a subgroup and provide various examples. We also consider cyclic subgroups and subgroups generated by subsets in a given group G. Example include A4 and D8. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-

From playlist Abstract Algebra

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Genevieve Walsh: Incoherence of free-by-free and surface-by-freegroups

CIRM VIRTUAL EVENT Recorded during the meeting"Virtual Geometric Group Theory conference " the May 29, 2020 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM

From playlist Virtual Conference

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Stability and Invariant Random Subgroups - Henry Bradford

Stability and Testability Topic: Stability and Invariant Random Subgroups Speaker: Henry Bradford Affiliation: Cambridge University Date: January 20, 2021 For more video please visit http://video.ias.edu

From playlist Stability and Testability

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Abstract Algebra class April 13, 2021

Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5 Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/michael-penn-math Personal Website: http://www.michael-penn.net Randolph College Math: http://www.randolphcollege.edu/ma

From playlist Super Lo-fi in class videos

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Angelica Babei - A family of $\phi$-congruence subgroups of the modular group

In this talk, we introduce families of subgroups of finite index in the modular group, generalizing the congruence subgroups. One source of such families is studying homomorphisms of the modular group into linear algebraic groups over finite fields. In particular, we examine a family of no

From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory

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Calculations with Matrix groups over the integers by Alexander Hulpke

DATE & TIME 05 November 2016 to 14 November 2016 VENUE Ramanujan Lecture Hall, ICTS Bangalore Computational techniques are of great help in dealing with substantial, otherwise intractable examples, possibly leading to further structural insights and the detection of patterns in many abstra

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Geometric structures and thin groups I - Alan Reid

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From playlist Mathematics

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Group theory 32: Subgroups of free groups

This lecture is part of an online mathematics course on group theory. We describe subgroups of free groups, show that they are free, calculate the number of generators, and give two examples.

From playlist Group theory

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Talk by Yanli Song in Global Noncommutative Geometry Seminar (Americas) http://www.math.wustl.edu/~xtang/NCG-Seminar.html on May 6, 2020.

From playlist Global Noncommutative Geometry Seminar (Americas)

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Tim Steger: ​Construction of lattices defining fake projective planes - lecture 8

Recording during the meeting "Ball Quotient Surfaces and Lattices " the March 01, 2019 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathe

From playlist Algebraic and Complex Geometry

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Inverse normal with Z Table

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From playlist Unit 2: Normal Distributions

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