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Does there exist a group $G$, a subgroup $N\le G$, and an automorphism $\alpha$ of $G$ such that $\alpha(N)\subset N$, but $\alpha$ does not restrict to an automorphism of $N$?

Equivalently, can there be subgroups of a group which are stable under an automorphism of the ambient group, but not stable under its inverse?

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  • $\begingroup$ Yes, when the group is infinite this can even happen for conjugation (there is a nice duplicate target with this case, but I can't find it right now). $\endgroup$ – Tobias Kildetoft Oct 12 '18 at 20:20
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    $\begingroup$ $G=\Bbb Q$, $N=\Bbb Z$, $\alpha(x)=2x$ $\endgroup$ – user8268 Oct 12 '18 at 20:29
  • $\begingroup$ @user8268 Ah, great. If you post your comment as an answer I can accept it. Thanks! $\endgroup$ – user355183 Oct 12 '18 at 20:34
  • $\begingroup$ It restricts ton an injective endomorphism of $N$. Conversely, every injective endomorphism $f$ of a group $N$ extends to an automorphism of some larger group, namely the inductive limit of $N\stackrel{f}\to N\stackrel{f}\to N\dots$ $\endgroup$ – YCor Oct 12 '18 at 21:16
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$G=\Bbb Q$, $N=\Bbb Z$, $\alpha(x)=2x$

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