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).
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
Definition of a Subgroup in Abstract Algebra with Examples of Subgroups
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From playlist Abstract Algebra
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
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
Percentiles, Deciles, Quartiles
Understanding percentiles, quartiles, and deciles through definitions and examples
From playlist Unit 1: Descriptive Statistics
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
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
Populations, Samples, Parameters, and Statistics
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From playlist Statistics
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
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
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
Abstract Algebra class April 13, 2021
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From playlist Super Lo-fi in class videos
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
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
From playlist Group Theory and Computational Methods
Geometric structures and thin groups I - Alan Reid
Speaker: Alan Reid (UT Austin/IAS) Title: Geometric structures and thin groups I Abstract: In these two talks we will discuss situations in which geometric input can be used as a method to certify that a group is thin. This involves a mix of theory and computation. More videos on http://v
From playlist Mathematics
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
Yanli Song: Higher index theorem for proper actions of Lie groups
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)
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
Determining values of a variable at a particular percentile in a normal distribution
From playlist Unit 2: Normal Distributions