Math 71 Homework Assignment 25 - 29, October 1999

1. p. 124: 8, 10, 14

2. Let G be a group of order 105. Show that G has both a normal Sylow 5- subgroup and a normal Sylow 7-subgroup.

3. Let G be a group of order 48. Show that G has a normal subgroup of order 8 or 16.

4. Let G be a group of order 231, and suppose that G has only one Sylow 3-subgroup. Show that G is cyclic. A few hints... For problem 2:

(a) First show that if H is a group of order 35, all its Sylow p-subgroup s are normal in H (i.e. n_{5} = n_{7} = 1).

(b) Next show that if G is a group of order 105, for at least one of p = 5 or p = 7, we have n_{p} = 1.

(c) For each p = 5, 7, let H_{p} denote a fixed Sylow p-subgroup of G. Show that H = H_{5}H_{7} is a normal subgroup of G.

(d) Let P be any Sylow p-subgroup of G, p = 5 or 7. Show that P = H_{p}. For problem 3:

(a) If there is more than one Sylow 2-subgroup, let H and K be any two of them. Show that |H ∩ K| = 8.

(b) Show that H, K ⊂ N_{G}(H ∩ K).

(c) Show that G = N_{G}(H ∩ K).