Let $G$ be a finite $p$-group. Suppose that $H,K$ are subgroups of $G$, such that $H \leq K$. If $\operatorname{Aut}(H)$ denotes the group of automorphism of $H$, then is there any relationship between $\operatorname{Aut}(H)$ and $\operatorname{Aut}(K)$? That is, may happen that $\operatorname{Aut}(H) \leq \operatorname{Aut}(K)?$ or $\operatorname{Aut}(K) \leq \operatorname{Aut}(H)?$ Or, in general, there is no relationship between these groups?
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1What is the role of $ G $? – Tobias Kildetoft Aug 08 '13 at 14:24
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$G$ is a any finite $p$-group – Aug 08 '13 at 14:42
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2Yes, but what role does it play. No part of the question seems to mention it. – Tobias Kildetoft Aug 08 '13 at 14:43
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The coice of $G$ may play a role for the answer. For example, take $G=\mathbb{Z}/p\mathbb{Z}$. – Dietrich Burde Aug 08 '13 at 15:21
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1If, for example, $H$ is a direct factor of $K$, then there is a natural embedding of ${\rm Aut}(H)$ in ${\rm Aut}(K)$, but there is not much more that you can say in general. – Derek Holt Aug 08 '13 at 15:32
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Also, if $H$ is characteristic in $K$ then there is a natural map from $\rm{Aut}(K)$ to $\rm{Aut}(H)$ given by restriction. But this map will rarely be injective (if it is, then this means that any automorphism that acts trivially on $H$ must also act trivially on $K$). – Tobias Kildetoft Aug 09 '13 at 06:45
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@DietrichBurde No, $G$ plays no role here other than to limit the possible choices of $H$ and $K$. But since $G$ can be any $p$-group, so can $H$ and $K$. – Tobias Kildetoft Aug 09 '13 at 06:47
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The dihedral group $D_8$ of order $8$ contains a copy of the Klein $4$-group $V \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. Now $\operatorname{Aut}(D_8) \cong D_8$ and $\operatorname{Aut}(V) \cong S_3$. So in this case $\operatorname{Aut}(D_8) \not\leq \operatorname{Aut}(V)$ and $\operatorname{Aut}(V) \not\leq \operatorname{Aut}(D_8)$ (by Lagrange's theorem), even though $V \leq D_8$.
Mikko Korhonen
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