# Abelian-potentially characteristic subgroup

From Groupprops

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup of an abelian group is termed an **abelian-potentially characteristic subgroup** if there exists an abelian group containing such that is a characteristic subgroup of .

## Relation with other properties

### Stronger properties

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

Characteristic subgroup of abelian group | characteristic subgroup of abelian group | direct from definition | abelian-potentially characteristic not implies characteristic | |FULL LIST, MORE INFO |

Abelian-potentially fully invariant subgroup | is a fully invariant subgroup in some bigger abelian group | follows from fully invariant implies characteristic | ? | |FULL LIST, MORE INFO |

Abelian-potentially verbal subgroup | is a verbal subgroup in some bigger abelian group | |FULL LIST, MORE INFO | ||

Subgroup of finite abelian group | finite abelian NPC theorem | Abelian-potentially verbal subgroup|FULL LIST, MORE INFO |

### Weaker properties

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

Abelian-extensible automorphism-invariant subgroup | invariant under all abelian-extensible automorphisms | abelian-potentially characteristic implies abelian-extensible automorphism-invariant | ? | |FULL LIST, MORE INFO |

Subgroup of abelian group | subgroup of an abelian group | Abelian-extensible automorphism-invariant subgroup|FULL LIST, MORE INFO |