Is it true that two isomorphic subgroups are in the same orbit (group acting on all its subgroups)?

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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?






abstract-algebra group-theory







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  • No. This would roughly say that all autos must be inner, which isn't so.
    – Randall
    1 hour ago










  • What autos and inner stands for? And what result are you citing?
    – Daniel Li
    1 hour ago










  • Doesn't matter, due to TomGrubb's suggestion. But, "automorphisms" and "inner automorphisms."
    – Randall
    1 hour ago














up vote
1
down vote

favorite












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?






abstract-algebra group-theory







share|cite|improve this question





















  • No. This would roughly say that all autos must be inner, which isn't so.
    – Randall
    1 hour ago










  • What autos and inner stands for? And what result are you citing?
    – Daniel Li
    1 hour ago










  • Doesn't matter, due to TomGrubb's suggestion. But, "automorphisms" and "inner automorphisms."
    – Randall
    1 hour ago












up vote
1
down vote

favorite









up vote
1
down vote

favorite











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?






abstract-algebra group-theory







share|cite|improve this question













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?






abstract-algebra group-theory




abstract-algebra group-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 1 hour ago









Daniel Li

508212




508212











  • No. This would roughly say that all autos must be inner, which isn't so.
    – Randall
    1 hour ago










  • What autos and inner stands for? And what result are you citing?
    – Daniel Li
    1 hour ago










  • Doesn't matter, due to TomGrubb's suggestion. But, "automorphisms" and "inner automorphisms."
    – Randall
    1 hour ago
















  • No. This would roughly say that all autos must be inner, which isn't so.
    – Randall
    1 hour ago










  • What autos and inner stands for? And what result are you citing?
    – Daniel Li
    1 hour ago










  • Doesn't matter, due to TomGrubb's suggestion. But, "automorphisms" and "inner automorphisms."
    – Randall
    1 hour ago















No. This would roughly say that all autos must be inner, which isn't so.
– Randall
1 hour ago




No. This would roughly say that all autos must be inner, which isn't so.
– Randall
1 hour ago












What autos and inner stands for? And what result are you citing?
– Daniel Li
1 hour ago




What autos and inner stands for? And what result are you citing?
– Daniel Li
1 hour ago












Doesn't matter, due to TomGrubb's suggestion. But, "automorphisms" and "inner automorphisms."
– Randall
1 hour ago




Doesn't matter, due to TomGrubb's suggestion. But, "automorphisms" and "inner automorphisms."
– Randall
1 hour ago










1 Answer
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Hint: Conjugation in abelian groups is trivial.






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  • LOL, amazingly simple.
    – Randall
    1 hour ago










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1 Answer
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active

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
6
down vote













Hint: Conjugation in abelian groups is trivial.






share|cite|improve this answer




















  • LOL, amazingly simple.
    – Randall
    1 hour ago














up vote
6
down vote













Hint: Conjugation in abelian groups is trivial.






share|cite|improve this answer




















  • LOL, amazingly simple.
    – Randall
    1 hour ago












up vote
6
down vote










up vote
6
down vote









Hint: Conjugation in abelian groups is trivial.






share|cite|improve this answer












Hint: Conjugation in abelian groups is trivial.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 1 hour ago









TomGrubb

10.5k11337




10.5k11337











  • LOL, amazingly simple.
    – Randall
    1 hour ago
















  • LOL, amazingly simple.
    – Randall
    1 hour ago















LOL, amazingly simple.
– Randall
1 hour ago




LOL, amazingly simple.
– Randall
1 hour ago

















 

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