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Suppose that there are two groups $G$ and $H$, and there is a group homomorphism function $f: G \rightarrow H$. If there is a group homomorphism function $z: H \rightarrow G$ that is not an inverse of $f$, can $G$ and $H$ be said to be in bijection?

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    Can you think of any examples using specific groups? That might help...2012-08-18
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    There is _always_ a group homomorphism (no need to add the word function) $z: H \rightarrow G$ sending everything to the identity of $G$, and it is never the inverse of $f$ (except in one case that you should be able to guess). This should make you cautious of the kind of statement you ask about.2012-08-18

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