Is there a way to classify all metabelian finite groups $G$, such that $ operatornameAut(G) cong G$?

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Is there a way to classify all metabelian finite groups $G$, such that $ operatornameAut(G) cong G$?



I know, that the trivial group is the only abelian group, that satisfies those condition. I also know two non-abelian groups that satisfy those conditions: $S_3$ and $D_4$, but do not know, if there are any other.



Any help will be appreciated.



EDIT: Now I also know, that $Hol(Z_n)$ satisfies those conditions for every natural $n$ that does not divide $4$. But still, is there anything else?




abstract-algebra group-theory finite-groups automorphism-group




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  • There are others, see here, and its linked MO-questions, but I do not know a classification.
    – Dietrich Burde
    Jul 23 at 13:20















up vote
2
down vote

favorite
1












Is there a way to classify all metabelian finite groups $G$, such that $ operatornameAut(G) cong G$?



I know, that the trivial group is the only abelian group, that satisfies those condition. I also know two non-abelian groups that satisfy those conditions: $S_3$ and $D_4$, but do not know, if there are any other.



Any help will be appreciated.



EDIT: Now I also know, that $Hol(Z_n)$ satisfies those conditions for every natural $n$ that does not divide $4$. But still, is there anything else?




abstract-algebra group-theory finite-groups automorphism-group




share|cite|improve this question





















  • There are others, see here, and its linked MO-questions, but I do not know a classification.
    – Dietrich Burde
    Jul 23 at 13:20













up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





Is there a way to classify all metabelian finite groups $G$, such that $ operatornameAut(G) cong G$?



I know, that the trivial group is the only abelian group, that satisfies those condition. I also know two non-abelian groups that satisfy those conditions: $S_3$ and $D_4$, but do not know, if there are any other.



Any help will be appreciated.



EDIT: Now I also know, that $Hol(Z_n)$ satisfies those conditions for every natural $n$ that does not divide $4$. But still, is there anything else?




abstract-algebra group-theory finite-groups automorphism-group




share|cite|improve this question













Is there a way to classify all metabelian finite groups $G$, such that $ operatornameAut(G) cong G$?



I know, that the trivial group is the only abelian group, that satisfies those condition. I also know two non-abelian groups that satisfy those conditions: $S_3$ and $D_4$, but do not know, if there are any other.



Any help will be appreciated.



EDIT: Now I also know, that $Hol(Z_n)$ satisfies those conditions for every natural $n$ that does not divide $4$. But still, is there anything else?





abstract-algebra group-theory finite-groups automorphism-group





share|cite|improve this question












share|cite|improve this question




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edited Aug 2 at 16:45
























asked Jul 23 at 9:20









Yanior Weg

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  • There are others, see here, and its linked MO-questions, but I do not know a classification.
    – Dietrich Burde
    Jul 23 at 13:20

















  • There are others, see here, and its linked MO-questions, but I do not know a classification.
    – Dietrich Burde
    Jul 23 at 13:20
















There are others, see here, and its linked MO-questions, but I do not know a classification.
– Dietrich Burde
Jul 23 at 13:20





There are others, see here, and its linked MO-questions, but I do not know a classification.
– Dietrich Burde
Jul 23 at 13:20
















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