HW due Fri Nov 24Mon Nov 27

1. Let G be a p-group (i.e. |G|=p^k for p some prime). Use the action of G on itself by conjugation to show that G has nontrivial center.

2. Let G > H > N be a chain of groups, where N is normal in G. If H/N is normal in G/N, show that H is normal in G.

3. Let G be a p-group. Show that G has a normal subgroup of order p. Then show that G has subgroups of every order p^j dividing |G|. (Hint: a better statement is true -- it has a chain of normal subgroups.)

4. Find a 3-Sylow subgroup of each of S₆, S₉, S₁₅, and S₂₇. (If it helps, they have nice descriptions as semidirect products.)

5. Let p divide |G|, where |G| < p². Show that G has a normal p-Sylow. (Analytic number theory question: what fraction of numbers |G| have such p? Answer: log 2, or most of them.)

6. Let p^a || |G|, q^b || |G|, where p,q are prime. Assume that the number of q-Sylows is not a multiple of p. Show that G has a subgroup of order p^a q^b.