Review of Group Theory: Cosets and Lagrange’s Theorem (Pt. II)
Point of post: This is a continuation of this post.
If and we call the set a left coset of in , or just a left coset when there is no ambiguity. Similarly we define a right coset of and denote it . We denote the set of all left cosets by , put more explicitly
and we denote by , this is called the index of in .
Remark: It’s clear that the number of left cosets of is equal to the number of right cosets of . This is because it’s easy to verify is a bijection between the two sets.
The first thing one notices about cosets is that:
Theorem: Let be a group and . Define the relation on by
then is an equivalence relation and (where denotes the equivalence class of under ).
Proof: We first prove that is an equivalence relation. To do this we note that clearly for every since . To prove that is symmetric we note that if then and thus so that . Lastly, to prove that is transitive we note that if and then and and thus . From these three axioms it follows that is an equivalence relation.
Next, to prove that it suffices to note that if then so that for some and thus for some and so . Conversely, if then for some and so . The conclusion follows.
Corollary: Let then
(where is meant to denote that the elements of are pairwise disjoint, and they’re union is all of . In other words, forms a partition of ).
Remark: Evidently the same methodology applies to show that the set of right cosets of partitions .
We are now ready to prove one of the main results of basic finite group theory
Theorem(Lagrange): Let be a finite group and and . Then:
: Recall that
But, we know from the first part of Cayley’s Theorem that (translation by ) is a bijection so that . Thus, it follows that
: This follows immediately from .
: This follows from previous theorem since and .
: This follows immediately from since
1. Lang, Serge. Undergraduate Algebra. 3rd. ed. Springer, 2010. Print.
2. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 200