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A finite group $G$ is metabelian if its commutator subgroup is abelian. Equivalently, $G$ is an extension of an abelian group by an abelian group. Metabelian groups are solvable, and they include all abelian, dihedral, quaternion, metacyclic, extra​special groups and groups of $p$-rank one.

Every metacylic group is metabelian, but the converse does not hold, as shown by these examples.

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  • Review status: reviewed
  • Last edited by David Roe on 2021-10-13 23:18:41
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