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Problem 98 (External Direct Products Of Abelian And Cyclic Groups)

Suppose that $G$ and $H$ are groups.

  1. Prove or disprove: If both $G$ and $H$ are Abelian, then $G\oplus H$ is Abelian.
  2. Prove or disprove: If $G\oplus H$ is Abelian,then both $G$ and $H$ are Abelian.
  3. Prove or disprove: If both $G$ and $H$ are cyclic, then $G\oplus H$ is cyclic.
  4. Prove or disprove: If $G\oplus H$ is cyclic, then both $G$ and $H$ are cyclic.
Hint: One of the four statements above is false. The rest are all true.

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