Cosets and Lagrange’s Theorem (10/28)

1. Cosets and Lagrange’s Theorem (10/28) • Definition. If H is a subgroup of G and if a G, thenthe 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, Andoes 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 everya 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 147-149. • Finish up Exercises 1-9 on page 156 and also do Exercises 14, 15, 16, 17, 18, 19, 22, and 23.