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Problem 97 (Images Of Abelian And Cyclic Groups)

Let $f:G\to H$ be a homomorphism. We have already shown that the image of $f$, written $f(G)$, is a subgroup of $H$.

  1. Prove that if $G$ is Abelian, then $f(G)$ is Abelian.
  2. Prove that if $G$ is cyclic, then $f(G)$ is cyclic.

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