Composition subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Definition

Symbol-free definition

A subgroup of a group is termed a composition subgroup if there is a composition series starting at the subgroup and ending at the whole group.

Relation with other properties

Stronger properties

• Sub-(maximal normal) subgroup: For full proof, refer: Sub-(maximal normal) implies composition

Weaker properties

• Serial subgroup: For full proof, refer: Composition implies serial

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Any composition subgroup of a composition subgroup is a composition subgroup -- we can concatenate the two composition series.