Subgroups (9/18)

1. Subgroups (9/18) • Definition. A subset H of a group G is called a subgroup of G if H is itself a group under the same operationas G. • To check that H is a subgroup of G, we must ask three questions: • Is H closedunder G’s operation? • Is e in H? • If a is any element in H, is a-1 also in H ?(“Closed under inverses”) • This is called the “Three Step Verification”. As you can read in the text, there are also (slicker) Two Step and One Step Verifications, but I prefer to simply always check the three key things.

2. Is H a subgroup of G ? • Click A for “yes”, B for “no” • G = Z, H = Z+ • G = Z, H = 2 Z • G = Z, H = nZ (n any integer) • G = Q, H = Q* • G = Q*, H = Q+ • G = Q*, H = { 2n | n  Z } • G = C, H = R • G = C*, H = R* • G = GL(4, R), H = GL(2, R) • G = GL(2, R), H = GL(2, Q) • G = GL(2, R), H = SL(2, R) • G = GL(2, R),H = {M  GL(2, R)| det(M) = 3n, n Z }

3. Is H a subgroup of G ? (Continued) • G = any group, H = {e} • G = any group, H= G • G = Z10, H= {0, 2, 4, 6, 8} • G = Z10, H = {0, 3, 6, 9} • G = Z10, H = U(10) • G = U(10), H = {1, 9} • G = Dn, H= {the set of all rotations} • G = Dn, H = {the set of all flips} • G = D4 , H = {R0, R180,H, V} • G = S4 , H = D4

4. Cyclic Subgroups • Here is a simple way to form subgroups of any group: • Definition. If a is any element of a group G, the cyclic subgroup of G generated by a,denoted a, is {an | n  Z}. • Example: In Q*, what is 5 ? • Example: In Z , what is 5 ? • Example: In D4 , what is R90 ? • So, in a finite group, we only need to look at positive powers of a, and we will eventually end up at the identity e = a0, and we start over again. This is where the term “cyclic” comes from! (Why do we have to cycle back??) • Example: In Z10, what is 6 ?

5. Assignment for Friday • Hand-in #1 is due. • Read Chapter 3. • On pages 68-69, do Exercises 1-10.