In mathematics, a metabelian group is a group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there is an abelian normal subgroup A such that the quotient group G/A is abelian.
Subgroups of metabelian groups are metabelian, as are images of metabelian groups over group homomorphisms.
- Any dihedral group is metabelian, as it has a cyclic normal subgroup of index 2. More generally, any generalized dihedral group is metabelian, as it has an abelian normal subgroup of index 2.
- If F is a field, the group of affine maps (where a ≠ 0) acting on F is metabelian. Here the abelian normal subgroup is the group of pure translations , and the abelian quotient group is isomorphic to the group of homotheties . If F is a finite field with q elements, this metabelian group is of order q(q − 1).
- The group of direct isometries of the Euclidean plane is metabelian. This is similar to the above example, as the elements are again affine maps. The translations of the plane form an abelian normal subgroup of the group, and the corresponding quotient is the circle group.
- The finite Heisenberg group H3,p of order p3 is metabelian. The same is true for any Heisenberg group defined over a ring (group of upper-triangular 3 × 3 matrices with entries in a commutative ring).
- All nilpotent groups of class 3 or less are metabelian.
- The lamplighter group is metabelian.
- All groups of order p5 are metabelian (for prime p).MSE
- All groups of order less than 24 are metabelian.
- Robinson, Derek J.S. (1996), A Course in the Theory of Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94461-6
- Ryan Wisnesky, Solvable groups (subsection Metabelian Groups)
- Groupprops, The Group Properties Wiki Metabelian group
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