nLab metabelian group

Metabelian groups

Metabelian groups


A metabelian or meta-abelian group is a group that is one step beyond (Greek ‘μετά’) being abelian. The steps here are those in the normal series? of a solvable group.


Let GG be a group. Then the following conditions are all equivalent:

  • GG satisfies the equational law? that every element of the form ababcdcdbabadcdca b a' b' c d c' d' b a b' a' d c d' c' is the identity element (where primes indicate inverses).

  • GG satisfies the equational law that [[a,b],[c,d]][[a,b],[c,d]] is the identity, where brackets indicate commutators.

  • The commutator subgroup of GG is abelian.

  • GG has an abelian subgroup which is normal and whose quotient is also abelian.

  • GG has an abelian quotient group whose kernel is also abelian.

  • GG is solvable with solvability at most 22.

In this case, we say that GG is meta-abelian, or metabelian for short.


The solvable groups with solvability exactly 22 are precisely the groups that are metabelian but not abelian. (Then the groups with solvability 11 are those that are abelian but not trivial, and the trivial group is the only group of solvability 00.)

Being metabelian is hereditary and cohereditary?. That is, subgroups and quotient groups of metabelian groups are also metabelian.


Every group of order less than 2424 is metabelian; the smallest non-metabelian group is the symmetric group S 4S_4.


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Last revised on August 5, 2018 at 09:20:29. See the history of this page for a list of all contributions to it.