Metabelian group

Symbol-free definition
A group is said to be metabelian or is said to have derived length two or solvable length two if it satisfies the following equivalent conditions:


 * 1) It is solvable of defining ingredient::derived length at most two, i.e., the derived series has length at most two.
 * 2) Its defining ingredient::derived subgroup (i.e., commutator subgroup with itself) is abelian.
 * 3) There is an defining ingredient::abelian normal subgroup that is also an defining ingredient::abelian-quotient subgroup, i.e., an abelian normal subgroup with abelian quotient group.
 * 4) Any two defining ingredient::commutators (i.e., elements that can be expressed as commutators of elements of the group) commute with each other.

Origin
The concept and term metabelian group was introduced by Furtwangler in 1930.

The term metabelian was earlier used for groups of nilpotency class two, which is a much stronger condition, but is no longer used in that sense.

Formalisms
The property of being metabelian arises by applying the meta operator to the group property of being Abelian. Equivalently metabelian can be described as Abelian-by-Abelian, where by denotes the group extension operator.

Generic examples

 * The trivial group is metabelian; in fact, it has derived length zero.
 * Any abelian group is metabelian; in fact, it has derived length zero.
 * Any group of nilpotency class two is metabelian.
 * Any group arising as the holomorph of a cyclic group (i.e., the semidirect product of a cyclic group and its automorphism group) is metabelian. This is because cyclic implies abelian automorphism group -- the automorphism group of a cyclic group is abelian. More generally, any group arising as the semidirect product of a cyclic group and any subgroup of its automorphism group is abelian.
 * Any dihedral group is metabelian: it has a cyclic normal subgroup and the quotient group is cyclic of order two. (This is a special case of the previous example). It turns out that dihedral groups are nilpotent only if their order is a power of two; further, the nilpotency class of a dihedral group of order $$2^n$$ is $$n - 1$$.
 * Any generalized dihedral group is metabelian: it has an abelian normal subgroup and the quotient group is cyclic of order two.

Particular examples

 * The symmetric group of degree three, which is also the dihedral group of order six and degree three, is metabelian. It is the smallest example of a metabelian non-abelian group; it is also not nilpotent. The abelian normal subgroup in this case is the alternating group, i.e., the subgroup generated by a $$3$$-cycle $$\langle (1,2,3) \rangle$$, and the quotient group is cyclic of order two.
 * The dihedral group of order eight (degree four) is metabelian. We have several choices here of the abelian normal subgroup. We can take the center as the abelian normal subgroup, in which case the quotient is isomorphic to a Klein four-group. Alternatively, we could take as our abelian normal subgroup any of the subgroups of order four.
 * the alternating group of degree four (order twelve) is metabelian. It has an abelian normal subgroup of order four: the Klein four-subgroup comprising the identity and the three double transpositions $$(1,2)(3,4), (1,3)(2,4), (1,4)(2,3)$$. The quotient is a cyclic group of order three.

Metaproperties
Any subgroup of a metabelian group is metabelian. This follows from the general fact that the derived series of the subgroup is contained (entry-wise) in the derived series of the whole group.

Any quotient of a metabelian group is metabelian. This follows from the fact that the derived series of the quotient is the quotient of the derived series of the original group.

A direct product of metabelian groups is metabelian. This follows from the fact that the derived series of the direct product is the direct product of the respective derived series.