Metabelian group explained

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. In fact, they are precisely the solvable groups of derived length at most 2.

Examples

x\mapstoax+b

(where a ≠ 0) acting on F is metabelian. Here the abelian normal subgroup is the group of pure translations

x\mapstox+b

, and the abelian quotient group is isomorphic to the group of homotheties

x\mapstoax

. If F is a finite field with q elements, this metabelian group is of order q(q − 1).

In contrast to this last example, the symmetric group S4 of order 24 is not metabelian, as its commutator subgroup is the non-abelian alternating group A4.

External links

Notes and References

  1. https://math.stackexchange.com/q/124010/178864 MSE