# Characteristic subgroup of group of prime power order

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property imposed on theambient group: group of prime power order

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Contents

## Definition

A **characteristic subgroup of group of prime power order** is a subgroup of a group where the group is a group of prime power order and the subgroup is a characteristic subgroup.

## Examples

Below are some examples of a proper nontrivial subgroup that satisfy the property characteristic subgroup in a group that satisfies the property group of prime power order.

Below are some examples of a proper nontrivial subgroup that *does not* satisfy the property characteristic subgroup in a group that satisfies the property group of prime power order.

Here is a more systematic look at the examples of order .

Group | Order | Characteristic subgroups | Non-characteristic subgroups |
---|---|---|---|

dihedral group:D8 (subgroup structure) | center of dihedral group:D8, cyclic maximal subgroup of dihedral group:D8, whole group, trivial subgroup | Klein four-subgroups of dihedral group:D8, non-central subgroups of order 2 | |

quaternion group (subgroup structure) | center of quaternion group, whole group, trivial subgroup | cyclic maximal subgroups of quaternion group | |

prime-cube order group:U(3,p) (subgroup structure) | center, trivial subgroup, whole group | non-central subgroups of order , subgroups (elementary abelian) of order | |

semidirect product of cyclic group of prime-square order and cyclic group of prime order (subgroup structure) | center, elementary abelian subgroup of order , trivial subgroup, whole group | non-central subgroups of order , cyclic subgroups of order |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Verbal subgroup of group of prime power order | (see also list of examples) | |||

Fully invariant subgroup of group of prime power order | (see also list of examples) | |||

Isomorph-free subgroup of group of prime power order | (see also list of examples) |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Finite-p-potentially characteristic subgroup | (see also list of examples) | |||

Normal subgroup of group of prime power order | (see also list of examples) | |||

Characteristic subgroup of finite group |

## Metaproperties

Metaproperty name | Satisfied? | Proof |
---|---|---|

transitive subgroup property | Yes | characteristicity is transitive |

strongly intersection-closed subgroup property | Yes | characteristicity is strongly intersection-closed |

strongly join-closed subgroup property | Yes | characteristicity is strongly join-closed |

intermediate subgroup condition | No | characteristicity does not satisfy intermediate subgroup condition (see examples with group of prime power order) |

transfer condition | No | (follows from statement for intermediate subgroup condition) |

image condition | No | characteristicity does not satisfy image condition (see examples with group of prime power order) |

centralizer-closed subgroup property | Yes | characteristicity is centralizer-closed |

commutator-closed subgroup property | Yes | characteristicity is commutator-closed |

upper join-closed subgroup property | No | characteristicity is not upper join-closed (See example with group of prime power order) |

## Effect of property operators

Operator | Meaning | Result of application | Proof |
---|---|---|---|

potentially operator | characteristic in some larger group of prime power order | finite-p-potentially characteristic subgroup | (by definition) |

### The potentially operator

*Applying the potentially operator to this property gives*: finite-p-potentially characteristic subgroup