7: The Wonderful World of Cosets

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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.