Centralizer-closed subgroup property
From Groupprops
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
Contents
Definition
A subgroup property is termed centralizer-closed if the following is true: whenever a subgroup
of a group
satisfies property
, so does the centralizer
.
Examples
Examples of subgroup properties that are centralizer-closed
Quick phrase | |
---|---|
Automorph-conjugate subgroup | |
C-closed normal subgroup | |
Characteristic subgroup | invariant under all automorphisms automorphism-invariant strongly normal normal under outer automorphisms |
Characteristic subgroup of group of prime power order | |
Finite direct power-closed characteristic subgroup |
Examples of subgroup properties that are not centralizer-closed
Quick phrase | |
---|---|
Subnormal subgroup | |
Transitively normal subgroup |