Group of finite composition length

Definition
A group is said to have finite composition length if it satisfies the following equivalent conditions:


 * 1) It possesses a defining ingredient::composition series of finite length, viz., a subnormal series such that all the successive quotients are defining ingredient::simple groups.
 * 2) Every defining ingredient::subnormal series (without repeated terms) can be refined to a composition series of finite length.
 * 3) It satisfies both the ascending chain condition on subnormal subgroups and the descending chain condition on subnormal subgroups.
 * 4) There is an upper bound on the length of any defining ingredient::subnormal series for the group (this upper bound equals the composition length, i.e., the length of any composition series).

Stronger properties

 * Weaker than::Finite group
 * Weaker than::Composition series-unique group
 * Weaker than::Composition factor-unique group
 * Weaker than::Composition factor-permutable group
 * Weaker than::Group whose chief series are composition series

Weaker properties

 * Stronger than::Group satisfying ascending chain condition on subnormal subgroups
 * Stronger than::Group satisfying descending chain condition on subnormal subgroups
 * Stronger than::Group satisfying ascending chain condition on normal subgroups
 * Stronger than::Group satisfying descending chain condition on normal subgroups
 * Stronger than::Group of finite chief length
 * Stronger than::Group in which all subnormal subgroups have a common bound on subnormal depth

Metaproperties
Any normal subgroup of a group of finite composition length again has finite composition length. In fact, if $$N$$ is a normal subgroup of a group $$G$$, the composition length of $$G$$ equals the sum of the composition lengths of $$N$$ and $$G/N$$.