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$\DeclareMathOperator{\Char}{char}$$\DeclareMathOperator{\Aut}{Aut}$ Let $G$ be a finite group, $H$ a subgroup, such that $H \Char G$. If $\phi \in \Aut(H)$ is there an automorphism $\widehat{\phi} \in \Aut(G)$ such that $\widehat{\phi}|_H$ = $\phi$?

In words, if $\phi$ is an automorphism of $H$, is there necessarily an automorphism $\widehat{\phi}$ of $G$ where $\widehat{\phi}$ maps the elements in $H$ to $H$ in exactly the same way?

I am thinking this is false (consider the non-finite example of $\bar{Q}$ and $\mathbb{C}$), but would like a counter example (for the finite case).

Soyuz
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  • Great question. I was wondering the same thing just a few days ago. I'd also be interested in any literature on this - in particular, when is it TRUE? – oxeimon Oct 13 '15 at 03:42
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    also I suspect this question would be okay with mathoverflow as well, as long as made it a bit more open ended (finite counterexamples are generally not that difficult to search for through GAP) – oxeimon Oct 13 '15 at 03:52
  • Good idea. As for when it is true, I believe it is true for any finite abelian group. – Soyuz Oct 13 '15 at 03:58
  • @oxeimon Here's the question on MO, FYI: http://mathoverflow.net/questions/220758/characteristic-subgroups-and-automorphisms – Soyuz Oct 13 '15 at 04:06
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    @Soyuz It is not true in the abelian group $C_4 \times C_2$ with $H = C_2 \times C_2$, which is a pretty small counterexample! – Derek Holt Oct 13 '15 at 08:21

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As you said, the answer is not always true, not always false.

Let $G$ be the non-abelian $p$-group ($p>2$), given by $$G=C_{p^2}\rtimes C_p=\langle x,y \colon x^{p^2}, y^p, yxy^{-1}=x^{1+p}\rangle .$$ In this group, $\langle x^p,y\rangle\cong C_p\times C_p$ is characteristic subgroup, since it is the only maximal subgroup which has exponent $p$. This characteristic subgroup has lot of automorphisms, take a simple one $$x^p\mapsto y, \mbox{ and } y\mapsto x^p.$$ This automorphism of $H$ can not be extended to an automorphism of $G$, because, $x^p$ is in center of $G$ but $y$ is not.

The example suggests that while looking extension of automorphisms from subgroup to group, we should also care about characteristic subgroups of $G$ which are inside $H$; such subgroups may not be characteristic in $H$, but in bigger group, they may become characteristic.

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