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$A,B,C$ are groups. Suppose that $A \times B \cong A \times C$ and both max-$n$ and min-$n$ hold in these groups. Prove that $ B \cong C$ (so that $A$ may be "cancelled"). Show that this condition is not generally valid.

The conditions max-$n$ and min-$n$ refer to the maximal and minimal conditions on normal subgroups respectively.

Let $\phi: A \times B \rightarrow A \times C$ be an isomorphism between the two product groups. I am considering to construct an automorphism of $A \times C$, which maps $\phi(1 \times B)$ onto $1 \times C$, or an automorphism of $A \times B$, mapping $1 \times B$ onto $\phi^{-1} (1 \times C)$. But I don't know how to apply the conditions...

Thank you very much!

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    @Shinya: I did not thing that you could *apply* Krull-Schmidt. Rather, my point is that this is essentially the base case in the induction for the uniqueness clause of the proof of the Krull-Schmidt Theorem. The ways in which I know how to prove this result all require some of the machinery that is built up to prove the K-S Theorem (though of course you do not need the full K-S to prove this). Hence my question as to what auxiliary results you might know, vs. what needs to be developed.2011-08-20

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