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

Let G be an abelian group and n a fixed positive integer. Prove that 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 = D_n for n ≥ 3 (i.e the statement above is false when G is not abelian).

Expert Solution

Colby Messinger answered 2 years ago

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