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If two subgroups are in the same orbit (with group acting on all of its subgroups by conjugation), they are isomorphic. But is the reverse true?

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  • $\begingroup$ No. This would roughly say that all autos must be inner, which isn't so. $\endgroup$
    – Randall
    Oct 2, 2018 at 2:59
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    $\begingroup$ What autos and inner stands for? And what result are you citing? $\endgroup$
    – Daniel Li
    Oct 2, 2018 at 3:02
  • $\begingroup$ Doesn't matter, due to TomGrubb's suggestion. But, "automorphisms" and "inner automorphisms." $\endgroup$
    – Randall
    Oct 2, 2018 at 3:04

1 Answer 1

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Hint: Conjugation in abelian groups is trivial.

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    $\begingroup$ LOL, amazingly simple. $\endgroup$
    – Randall
    Oct 2, 2018 at 3:02
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    $\begingroup$ A non-abelian example is $S_3\times\mathbb{Z}_2$, for a similar reason. $\endgroup$
    – user1729
    Oct 2, 2018 at 13:43

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