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Let G be a topological group and a be an element of order 2 in G. Further suppose the element a does not belong to center of G. Then is it true that only homeomorphism f of G such that $f(ax)=f(x)a$ for all x in G is $f(x)=x^{-1}$.

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No; the simplest counterexample is $S_{3}$ with the discrete topology. –  Sam Nolen Sep 4 '11 at 4:36
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1 Answer 1

The answer is no. Take $H$ any topological group, $H'$ another topological group having a noncentral element $a'$ of order 2, $G=H\times H'$, $a=1_H\times a'$ and $f=h\times inv_{H'}$ where $h$ is any anti-automorphism of $H$ and $inv_{h'}$ is the map $h'\mapsto h'^{-1},h'\in H'$.

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