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. Metabelian groups are solvable.
Examples
- 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 Fq is a finite field with q elements, the group of affine maps <math> x \mapsto ax+b </math> (where a ≠ 0) acting on Fq is metabelian. Here the abelian normal subgroup is the group of pure translations <math> x\mapsto x+b </math> (a group of order q ), its abelian quotient group is isomorphic to the group of homotheties <math> x\mapsto ax </math> (a cyclic group of order q − 1 ).
- The finite Heisenberg group H3,p of order p3 (see the third example Heisenberg group modulo p in the examples section) 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).
- The symmetric group on four letters S4 is solvable but is not metabelian because its commutator subgroup is the alternating group A4 which is not abelian.
External links
- Ryan J. Wisnesky, Solvable groups (subsection Metabelian Groups)
