set G be a finite group, P E Syl,(G), and N = (a) Show that Pe Syl, (N). Jumber and Conjugacy of Sylow Subgroups* 153 (b) Show that the normalizer in N of P is N. In other words, NN(P) = N. (c) Show that P is the unique Sylow p-subgroup of N. In other words, Syl, (N)| = 1.

I need help with number 4

A group G has come to talk to you. You find out that G has 100 = 22 x 52 elements and that G has exactly one subgroup of order 10 whose name is Н. (a) Remind yourself, by rereading Example 10.7, why H must be a nor- mal subgroup of G. (b) Let PE Syl;(G). What can you say about |P|, |H^P|, and |(H, P)|? Prove your assertions. (c) The group G is guaranteed to have subgroups of which sizes?

Suppose a cyclic group G = <a> with the older 12: (1). List all subgroups of G (2). List all those subgroups' generators