# Difference between revisions of "Characteristic subgroup of finite group"

From Groupprops

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− | ! | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions |

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− | | [[Weaker than:: | + | | [[Weaker than::isomorph-free subgroup of finite group]] || || || || |

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− | | [[Weaker than:: | + | | [[Weaker than::fully invariant subgroup of finite group]] || || || || |

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− | | [[Weaker than:: | + | | [[Weaker than::normal Sylow subgroup]] || || || || |

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− | | [[Weaker than:: | + | | [[Weaker than::normal Hall subgroup]] || || || || |

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− | | [[Weaker than:: | + | | [[Weaker than::xharacteristic subgroup of group of prime power order]] || || || || |

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! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions | ! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions | ||

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− | | [[Stronger than:: | + | | [[Stronger than::normal subgroup of finite group]] || || || || |

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− | | [[Stronger than:: | + | | [[Stronger than::finite characteristic subgroup]] || || || || |

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− | | [[Stronger than:: | + | | [[Stronger than::finite normal subgroup]] || || || || |

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## Revision as of 14:31, 1 June 2020

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

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 subgroup of a group is termed a **characteristic subgroup of finite group** if it satisfies the following equivalent conditions:

- The whole group is a finite group and the subgroup is a characteristic subgroup of it.
- The whole group is a finite group and the subgroup is a strictly characteristic subgroup (i.e., invariant under all surjective endomorphisms) of it.
- The whole group is a finite group and the subgroup is an injective endomorphism-invariant subgroup of it.
- The whole group is a finite group and the subgroup is a purely definable subgroup of it, i.e,, the subgroup is definable in the first-order theory of the group.
- The whole group is a finite group and the subgroup is an elementarily characteristic subgroup of it, i.e., there is no other elementarily equivalently embedded subgroup.
- The whole group is a finite group and the subgroup is a monadic second-order characteristic subgroup of it.

## Relation with other properties

### Stronger properties

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

isomorph-free subgroup of finite group | ||||

fully invariant subgroup of finite group | ||||

normal Sylow subgroup | ||||

normal Hall subgroup | ||||

xharacteristic subgroup of group of prime power order |

### Weaker properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

normal subgroup of finite group | ||||

finite characteristic subgroup | ||||

finite normal subgroup |