# Example of fixed point free of order 2 which is not inverse mapping

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 $$\phi (x)= x^{-1}$$.

• 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 $\pm1$ – Hagen von Eitzen May 28 at 20:39
• math.stackexchange.com/questions/858078/… – Believer May 28 at 20:49
• 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 – Hagen von Eitzen May 28 at 21:04