$A,B,C$ 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!