A problem in Isaacs' Algebra states that if a finite group has an involution (automorphism) that fixes only the identity, then that group is necessarily Abelian. Does anyone know of a counter example in the infinite case? That is, a group as in the title of this question - an infinite non-Abelian group with an involutive automorphism that preserves only the identity.
(This is Problem 2.3)
The free group on two generators $a,b$ and the involution is given by $a\mapsto b$, $b\mapsto a$.