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Let G be an abelian group and n a fixed positive integer. Prove that the following...

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Advance Math

Let G be an abelian group and n a fixed positive integer. Provethat the following sets are subgroups of G.

(a) P(G, n) = {gn | g ? G}.

(b) T(G, n) = {g ? G | gn = 1}.

(c) Compute P(G, 2) and T(G, 2) if G = C8 ×C2.

(d) Prove that T(G, 2) is not a subgroup of G = Dnfor n ? 3 (i.e the statement above is false when G is notabelian).

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