nLab metabelian group

Idea

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.

Definition

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

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

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

• The commutator subgroup of $G$ is abelian.

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

• $G$ has an abelian quotient group whose kernel is also abelian.

• $G$ is solvable with solvability at most $2$.

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

Properties

The solvable groups with solvability exactly $2$ are precisely the groups that are metabelian but not abelian. (Then the groups with solvability $1$ are those that are abelian but not trivial, and the trivial group is the only group of solvability $0$.)

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

Examples

Every group of order less than $24$ is metabelian; the smallest non-metabelian group is the symmetric group $S_4$.

References

• English Wikipedia. web.