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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)

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up vote 6 down vote accepted

The free group on two generators $a,b$ and the involution is given by $a\mapsto b$, $b\mapsto a$.

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