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7: The Wonderful World of Cosets

Last updated

Sep 15, 2021
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- 6.6: Exercises
- 7.1: Partitions of and Equivalence Relations on Sets

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We have already seen one way we can examine a complicated group $G$ : namely, study its subgroups, whose group structures are in some cases much more directly understandable than the structure of $G$ itself. But if $H$ is a subgroup of a group $G$ , if we only study $H$ we lose all the information about $G$ 's structure “outside” of $H$ . We might hope that $G−H$ (that is, the set of elements of $G$ that are not in $H$ ) is also a subgroup of $G$ , but we immediately see that cannot be the case since the identity element of $G$ must be in $H$ , and $H∩(G−H)=∅$ . Instead, let's ask how we can get at some understanding of $G$ 's entire structure using a subgroup $H$ ? It turns out we use what are called cosets of $H$ ; but before we get to those, we need to cover some preliminary material.

7.1: Partitions of and Equivalence Relations on Sets
The number of partitions of a finite set of n elements gets large very quickly as n goes to infinity. Indeed, there are 52 partitions of a set containing just 5 elements! (The total number of partitions of an n-element set is the Bell number. There is no trivial way of computing Bell number, in general, though the Bell number do satisfy the relatively simple recurrence relation.
7.2: Introduction to Cosets and Normal Subgroups
In this section, we will introduce cosets and normal subgroups, as well as providing the corresponding theorems and examples.
7.3: The Index of a Subgroup and Lagrange's Theorem
In this section, we discuss the index of a subgroup and Lagrange's Theorem, as well two related corollaries.
7.4: Exercises
This page contains the exercises for Chapter 7.

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