## Review of Group Theory: Cosets and Lagrange’s Theorem

**Point of post: **In this post we give a brief overview of the idea of cosets, normal subgroups, inner automorphisms, etc. Remember, this is just a perfunctory run through to get us in the mood for our rep. theory talk.

**Motivation**

It turns out that just as was the case with cosets of vector spaces, cosets of groups (defined below) play an important role in our theory. We will then see a special case of how cosets of a subgroup carve up the ambient group in a nice way which will give us much information about when and when not a subgroup can be a group. In particular, we prove Lagrange’s theorem which states that (roughly) the order of a subgroup of a finite group must divide the order of the ambient group.

*The Product of Subsets of a Group*

In what follows it will be useful to define the *product* of two subsets of a group . Explicitly if then we define to be the set

If we denote by and similarly if . Some basic theorems which can be derived about the product of subsets is

**Theorem: ***Let be a group and . Then,*

**Proof: **

: This is clear since

$latex\ mathbf{(2)}$: Similarly,

: Just as before

: This follows since if then and since is closed under multiplication it follows that . Conversely, if and then and so . The conclusion follows.

The last question we address is precisely when given two subgroups is ?

**Theorem: ***Let be a group and . Then, if and only if .*

**Proof: **Suppose first that and let . Then, and . Thus,

But, note that if we label then we may rewrite the above as

But, since and we know that for some and . Thus, we have that

but, if we label then we get that

Thus, since we may conclude by previous theorem that .

Conversely, suppose that and let then since we know that and thus for some and . Thus,

Conversely, we note that since and that . The conclusion follows.

**References:**

1. Lang, Serge. *Undergraduate Algebra*. 3rd. ed. Springer, 2010. Print.

2. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 200

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