Commutator-closed subgroup property
From Groupprops
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions
Definition
Symbol-free definition
A subgroup property is termed commutator-closed if the commutator of any two subgroups, each of which has property
in the whole group, also has property
in the whole group.
Definition with symbols
A subgroup property is termed commutator-closed if, given any group
and subgroups
such that
and
both satisfy property
in
, the commutator
also satisfies property
in
.
Relation with other metaproperties
Stronger metaproperties
- Endo-invariance property: For full proof, refer: Endo-invariance implies commutator-closed