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Questions tagged [metabelian-groups]

A metabelian group is a group, that is an extension of an abelian group by an abelian group. To be used with the tag [group-theory]

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Is a finite centerless metabelian group always a semidirect product of two abelian groups?

Suppose $G$ is a finite centerless metabelian group. Is it true that it is a semidirect product of two abelian groups? It does not seem true to me, but I failed to find any counterexamples. Actually, ...
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Is there a way to classify all metabelian finite groups $G$, such that $ \operatorname{Aut}(G) \cong G$?

Is there a way to classify all metabelian finite groups $G$, such that $ \operatorname{Aut}(G) \cong G$? I know that the trivial group is the only abelian group that satisfies this condition. I also ...