🚀 go-pugleaf

1.  prove that if Z(G) is the center of G, then Z(G) is a normal
subgroup of G.

2.  If G is a group and N is a normal subgroup of G , such that G/N is
abelian, prove that aba^-1b^-1 belong to N for all a, b belong to G.
(a^-1  mean a inverse same as b^-1)

3.  Let G be a group, H a subgroup of G, and N is a normal subgroup of
G.  Let the set HN = {hn | h belong to H, n belong to N}. Prove that :

(a)  H intersect N is a normal subgroup of H.
(b)  HN is a subgroup or G.
(c)  N is a subgroup of HN and N is a normal subgroup of HN.
(HN) / N  and H / (H intersection N) are isomorphic