# Can an automorphism of a group which stabilizes a subgroup fail to restrict to an automorphism of the subgroup?

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?

• 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). – Tobias Kildetoft Oct 12 '18 at 20:20
• $G=\Bbb Q$, $N=\Bbb Z$, $\alpha(x)=2x$ – user8268 Oct 12 '18 at 20:29
• @user8268 Ah, great. If you post your comment as an answer I can accept it. Thanks! – user355183 Oct 12 '18 at 20:34
• 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 $N\stackrel{f}\to N\stackrel{f}\to N\dots$ – YCor Oct 12 '18 at 21:16

$$G=\Bbb Q$$, $$N=\Bbb Z$$, $$\alpha(x)=2x$$