Let $G$ to be a infinite abelian 2-group. I want to know information in structure of group automorphism of $G$. For example $\alpha$ such that $\alpha(x)=x^{-1}$ for all $x\in G$ is an automorphism of $G$. Do we can find other automorphisms of $G$?
On automorphisms of infinite abelian 2-groups
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group-theory
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1In general? Depends on your set theory. There are set theories in which there e$x$ist infinite abelian groups of exponent $2$ with no non-trivial automorphisms (see [this](http://math.stackexchange.com/questions/281$4$5/axiom-of-choice-and-automorphisms-of-vector-spaces)). The automorphism you give would be the identity in that case. – 2012-01-28