A normal subgroup h lt g is a maximal normal subgroup of g

Question

(1) Prove that if Z(G) < G, then G/Z(G) is not cyclic.

(2) Let G be a group with Ng G.

(a) Prove that Z(GIN)= (Ng I [g, E N, ho E G}.

(b) Prove that (GIN)' = NG' IN .

(This generalizes the result that GIN is abelian if and only if G' < N.)

(3) Given an example to show that if H < G then G need not contain a subgroup iso­morphic to G/H.

(4) A normal subgroup H < G is a maximal normal subgroup of G if there is no normal subgroup If of G such that H < K that H is a maximal normal subgroup if and only if GI H is simple.

(5) Suppose JSHx K. Must J have the form J=AxB for some A < H and some B < K?

(6) Let G be a group.

(a)  Prove, for every G,o E G, that C,(axa^-')= aC_G(x)G^-'^-.

(b)  Prove that if H < G and h E H, then CH(h)= CG(h) n H.

(7) Let G be a group.

(a) Prove, for every a E G, H     that N,(GHG^-')= aN,(H)a^-'.

(13) Prove that if H < K             then NK(H)= NG(H)n K.