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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 $G$?

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In general? Depends on your set theory. There are set theories in which there exist infinite abelian groups of exponent $2$ with no non-trivial automorphisms (see this). The automorphism you give would be the identity in that case. – Arturo Magidin Jan 28 '12 at 22:46

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