Composition subgroup
From Groupprops
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]
Contents
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.