metabelian

Metabelian is a term used in group theory and Lie theory to describe a specific level of commutativity in a structure. In group theory, a group G is metabelian if its commutator subgroup G' = [G,G] is abelian. Equivalently, the derived series G^(0) = G, G^(1) = G', G^(n+1) = [G^(n), G^(n)] terminates after two steps, so G'' = 1 and the derived length is at most 2. In particular, the quotient G/G' is abelian, and G is built from two abelian layers.

In the context of Lie algebras, a Lie algebra L is metabelian if its derived subalgebra [L,L] is abelian; equivalently [ [L,L], [L,L] ] = 0. Metabelian Lie algebras are a natural analogue of metabelian groups, capturing the idea that the noncommutativity of the algebra is confined to a single, abelian layer.

Examples and construction. Every abelian group or Lie algebra is metabelian. If G = A ⋊ B is a semidirect product of two abelian groups A and B, then G is metabelian because G' lies in A and A is abelian. The Heisenberg group, a non-abelian group of upper triangular 3×3 integer matrices with ones on the diagonal, is a classic metabelian group: its commutator subgroup equals its center, which is abelian. The corresponding Heisenberg Lie algebra is likewise metabelian.

Relation to solvability. Metabelian groups are solvable of derived length at most 2, but not every solvable group is metabelian; for example, the symmetric group S4 has derived length 3. Metabelian structures provide a useful intermediate level of complexity between abelian and general non-abelian objects.