Characteristicity is centralizer-closed
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., centralizer-closed subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about characteristic subgroup |Get facts that use property satisfaction of characteristic subgroup | Get facts that use property satisfaction of characteristic subgroup|Get more facts about centralizer-closed subgroup property
Contents
Statement
Property-theoretic statement
The subgroup property of being characteristic satisfies the subgroup metaproperty of being centralizer-closed.
Verbal statement
The centralizer of a characteristic subgroup is characteristic.
Statement with symbols
Suppose is a group and
is a characteristic subgroup of
. Then, the centralizer
of
in
is also a characteristic subgroup of
.
Generalizations
Auto-invariance implies centralizer-closed: Any subgroup property that can be described as the invariance property with respect to a certain automorphism property, is closed under taking centralizers.