# 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 metapropertyVIEW 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 | |
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Automorph-conjugate subgroup | |

C-closed normal subgroup | |

Central factor | factor in central product product with centralizer is whole group quotient action by outer automorphisms is trivial every inner automorphism restricts to an inner automorphism |

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 | |

Local powering-invariant subgroup | |

Normal subgroup | invariant under inner automorphisms, self-conjugate subgroup same left and right cosets kernel of a homomorphism subgroup that is a union of conjugacy classes |

Powering-invariant subgroup |

### Examples of subgroup properties that are not centralizer-closed

Quick phrase | |
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Subnormal subgroup | |

Transitively normal subgroup |