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Suppose that $G\cong H$ are isomorphic groups whose non-identity elements are all of order 2. Fix $g \in G$ and $h \in H$ with $g\neq 1$ and $h\neq 1$. My question is does there exist an isomorphism from $G$ to $H$ taking $g$ to $h?$

My guess is that we can construct such an isomorphism from the given one, but I haven't quite been able to make this work, so maybe it is not true?

user544795
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  • If the groups are finite, yes; if the groups are infinite and you assume the Axiom of Choice, yes; if the groups are infinite and you don’t assume the Axiom of Choice, then I don’t know but suspect the answer will be “no”. – Arturo Magidin Nov 17 '19 at 08:02
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    As a hint: such a group is a vector space over the field with two elements. – Arturo Magidin Nov 17 '19 at 08:02
  • I suppose the groups are finite. Then see here how to construct $G\cong C_2^n\cong H$ with $g\mapsto h$. – Dietrich Burde Nov 17 '19 at 09:03

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