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I just learn about fixed point free of order 2 and when G is finite, this is nothing but map $\phi (x)= x^{-1}$. I am looking for example of fixed point free of order 2 which is not the map which $\phi (x)= x^{-1}$.

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  • $\begingroup$ What do you mean with fixed point free of order 2, and with finite case? Your $\phi$ is of order 2 (aka, involutory), but has fixed points $\pm1$ $\endgroup$ – Hagen von Eitzen May 28 at 20:39
  • $\begingroup$ math.stackexchange.com/questions/858078/… $\endgroup$ – Believer May 28 at 20:49
  • $\begingroup$ Oh, so from Believer's comment I infer that a lot of context is missing: The question is about automorphisms, not maps, the domain is a group (abelian), and the trivial fixpoint $1$ is not counted $\endgroup$ – Hagen von Eitzen May 28 at 21:04

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