Any proper subgroup S of a nilpotent group G is a proper subgroup of its normalizer N_{G}(S) in G.

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?

Let G be the cyclic group Z_4 under addition, of integers modulo 4. Let H be the subgroup of multiples of 2, {0,2}. Write out the two distinct cosets of H. One is 0+H=H. The other is some g+H. Then write out the addition table for the quotient group G/H consisting of those two cosets. The sum of a+H and b+H will be a+b+H.