1. Cosets and Lagrange’s Theorem (10/28)
Definition. If H is a subgroup of G and if a  G, then the left coset of H containing a, denoted aH, is simply {ah | h  H}. Parallel definition for right coset.
If G is abelian, we can just say “coset”. Also, if G is an additive group, we write a + H.
Note that H (= eH) itself is always one of its cosets.
Example. In Z, what are the cosets of H = 5Z?
Example. In Z12, what are the cosets of H = 4?
Example. In D4, what are the left cosets of V? What are the right cosets of V?
Example. In D4, what are the left cosets of R180? What are the right cosets of R180?

2. Key Role of Cosets
Theorem. The cosets (left or right) of H in G always “partition” G, i.e., they are pairwise-disjoint, and their union is all of G.
Theorem. If H is finite, then |aH| = |H| for all a  G.
So the cosets of H :
cover all of G,
never overlap with each other,
and all have the same order.
Check these on previous examples.
This leads us to one of the central results of finite group theory:

3. Lagrange’s Theorem
Theorem. If G is a finite group and if H is a subgroup of G, then |H| divides |G|.
For what class of groups did we already know this to be true?
Note that if G and H are as in the theorem, then |G| / |H| is just the number of cosets of H in G.
More generally (since the following definition can apply to infinite groups also) the index of H in G, denoted [G:H], is the number of cosets of H in G.
Example. What is the index of 5Z in Z?
Example. What is the index of V in D4?

4. Is the converse of Lagrange true?
Theorem???? If the number m divides |G|, does G then have a subgroup H of order m?
Again, we know this (and more!) to be true about a particular class of groups, right?
But, alas, it is not true in general. A4 provides a counter-example, and in fact is the smallest group to do so.
In general, it turns out that for n > 3, An does not contain a subgroup of order n! / 4.
By a more advanced set of theorems, the Sylow Theorems (Chapter 24), if m = pk, i.e., if m is a power of a prime, then the Lagrange converse does hold.

5. Nice corollaries of Lagrange
Theorem. If a  G, |a| divided |G|.
Theorem. Every group of prime order is cyclic.
Theorem. If a  G, a|G| = e.
Fermat’s Little Theorem. For every a  Z and every prime p, a p mod p = a mod p.
Example. What is 540 mod 37?
We prove Fermat’s Little Theorem in MA 214 (Number Theory), but the proof is a little tricky and we can’t use this proof since “we” don’t know any group theory there.

6. Assignment for Wednesday
Read pages
Finish up Exercises 1-9 on page 156 and also do Exercises 14, 15, 16, 17, 18, 19, 22, and 23.