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need help in abstract algebra
Posted:
Oct 15, 1996 3:27 AM


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



