22. The center Z(G) of a group G is defined as Z(G) = = {a Є G|ax = xa for all x Є G}.

15. Prove that if for all in the group , then is abelian.

Find two groups of order 6 that are not isomorphic.

Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.

9. Find all homomorphic images of the octic group.

Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.

Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.

Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40

Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.

18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.