Suppose that $\mathbb Z_p \times A \simeq \mathbb Z_p \times B$, where $p$ is prime. Is it true, that $A \simeq B$?
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If $A,B$ are abelian, the answer is yes. See Gone's answer here: http://math.stackexchange.com/questions/2193/cancellation-of-direct-products – Apr 18 '13 at 03:09
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Thank you, it is very interesting. But it is not use, that $p$ is prime. – Alex-omsk Apr 18 '13 at 03:34
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@Alex-mosk, so is there any restriction about $A,B$ at all? It would be good to know the context of your question. – Apr 18 '13 at 03:43
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This is true with no assumptions on $A, B$. See the link. – Qiaochu Yuan Apr 18 '13 at 03:56