Metabelian group
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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 (it has a cyclic normal subgroup of index two).
- If Fq is a finite field with q elements, the group of affine maps [ x \mapsto ax+b ] (where a ≠ 0) acting on Fq is metabelian. Here the abelian normal subgroup is the group of pure translations [ x\mapsto x+b ] (a group of order q ), its abelian quotient group is isomorphic to the group of homotheties [ x\mapsto ax ] (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)
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