Composition series-unique group

Symbol-free definition
A group is said to be composition series-unique if it has a unique defining ingredient::composition series. In the case of a finite group, this can be expressed inductively as saying that that group is one-headed and that the head (viz the unique maximal normal subgroup) is itself composition series-unique.

Stronger properties

 * Simple group
 * Cyclic group of prime power order

Weaker properties

 * Stronger than::Composition factor-unique group
 * Stronger than::Group of finite composition length

Opposite properties

 * Composition factor-permutable group