# Group in which every subgroup of finite index has finitely many automorphic subgroups

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

### Equivalent definition in tabular format

No. | Shorthand | A group is a group in which every subgroup of finite index has finitely many automorphic subgroups if ... |
---|---|---|

1 | finitely many automorphic subgroups | for every subgroup of finite index in , there are only finitely many automorphic subgroups to in , i.e., there are only finitely many subgroups of for which there exists an automorphism of such that . |

2 | characteristic core has finite index | for every subgroup of finite index in , the characteristic core of in has finite index in . |

3 | contains characteristic subgroup of finite index | for every subgroup of finite index in , there exists a characteristic subgroup of finite index in such that is contained in . |

4 | normal subgroup, finitely many automorphic subgroups | for every normal subgroup of finite index in , there are only finitely many automorphic subgroups to in , i.e., there are only finitely many subgroups of for which there exists an automorphism of such that . |

5 | normal subgroup, characteristic core of finite index | for every normal subgroup of finite index in , the characteristic core of in has finite index in . |

6 | normal subgroup contains characteristic subgroup of finite index | for every normal subgroup of finite index in , there exists a characteristic subgroup of finite index in such that is contained in . |

## Relation with other properties

### Stronger properties

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

finite group | Group with finitely many homomorphisms to any finite group|FULL LIST, MORE INFO | |||

finitely generated group | finitely generated implies every subgroup of finite index has finitely many automorphic subgroups | Group with finitely many homomorphisms to any finite group|FULL LIST, MORE INFO | ||

group with finitely many homomorphisms to any finite group | finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups | every subgroup of finite index has finitely many automorphic subgroups not implies finitely many homomorphisms to any finite group | |FULL LIST, MORE INFO | |

simple group | (via finitely many homomorphisms to any finite group) | |||

group of finite composition length | (via finitely many homomorphisms to any finite group) |