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I'm trying to find group G s.t every subgroup of G is invariant under all automorphisms, or conditions for G. For example; cyclic groups and simple groups have this condition.

negar
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  • Simple groups dont't in general. For exsmple $A_5$ contains a cyclic $5$-group that is not characteristic. – Hagen von Eitzen Oct 18 '13 at 16:15
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    A finite such group must be the product of its Sylow groups. Now try to classify all $p$-groups having your property. – Hagen von Eitzen Oct 18 '13 at 16:39
  • If all subgroups are normal (= invariant under the inner automorphisms), the group $G$ is called Dedekind group. You should be able to use the info in the wikipedia to reduce your problem to the abelian case, which should be doable. I'd guess that only cyclic groups fulfill your criterion. – j.p. Mar 31 '15 at 09:44

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