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If $H$ is a subgroup of $G$, whether an automorphism of $H$ can always be extended to an automorphism of $G$, inducing an embedding of ${\rm Aut}(H)$ into ${\rm Aut}(G)$?

If not, what are some counter-examples? Under which conditions on $G$ and $H$, it is true?

Jacob Manaker
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Dey
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    This question needs some interpretation. An automorphism of $H$ isn't an automorphism of $G$, as it has a different domain, so the group of automorphisms of $H$ isn't even a subset of the group of automorphisms of $G$, let alone a subgroup. So, what are you really asking? – Gerry Myerson Mar 27 '22 at 06:21
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    While you're thinking about it, you might want to look at https://math.stackexchange.com/questions/1477524/characteristic-subgroups-and-automorphisms and https://math.stackexchange.com/questions/1478384/characteristic-subgroups-and-automorphisms and https://math.stackexchange.com/questions/3060195/on-automorphisms-of-groups-which-extend-as-automorphisms-to-every-larger-group and https://math.stackexchange.com/questions/100687/on-group-automorphism-of-subgroup-a-group-g (which is an exact duplicate of part of your question). – Gerry Myerson Mar 27 '22 at 06:21
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    @GerryMyerson: thanks a lot for the references. I understand your point. What I mean is when an automorphism of H can be extended to an automorphism of G. – Dey Mar 27 '22 at 07:02
  • @fitzcarraldo: Thanks for your help! Let h ∈ H, and g ∈ G \ H. Then g.h ∈ G \ H. Hence, ′(g.h) = g.h . But, ′(g.h) = ′(g). ′(h) = g. (h) . This implies: (h) = h for all h∈ H, which may not be the case. So I guess the ′ you defined would not work. – Dey Mar 27 '22 at 07:10
  • You are right, it doesn't work. –  Mar 27 '22 at 07:14
  • Another earlier question: link. For an example, consider $G = D_8$, $H = C_2 \times C_2$. – Mikko Korhonen Mar 27 '22 at 07:38
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    You should edit your post, so it says what you mean, clearly and unambiguously. – Gerry Myerson Mar 27 '22 at 11:10
  • Please ask one question at a time. – Shaun Mar 27 '22 at 12:13
  • @GerryMyerson: Done. – Dey Mar 27 '22 at 18:04
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    @Shaun: Done. Thanks for the suggestion; I will post the second part as a separate question. – Dey Mar 27 '22 at 18:06
  • Good. But the title still has it wrong. – Gerry Myerson Mar 27 '22 at 20:59
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    I’m voting to close this question because OP refuses to correct the title. – Gerry Myerson Mar 30 '22 at 05:23
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    @GerryMyerson: You can always change it yourself…. – Jacob Manaker Mar 30 '22 at 17:13
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    @Jacob, sure, I could. But I don't think it's too much to ask of people posting here asking for help, that when an error is pointed out to them, that they fix it themselves. – Gerry Myerson Mar 30 '22 at 21:58
  • So, Dey, have the links users have posted helped you to see an answer to your question? – Gerry Myerson Mar 31 '22 at 10:58

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