# Group property-conditionally potentially characteristic subgroup

From Groupprops

This term is related to: potentially characteristic subgroups characterization problem

View other terms related to potentially characteristic subgroups characterization problem | View facts related to potentially characteristic subgroups characterization problem

## Definition

Suppose is a group property. Suppose is a group satisfying and is a subgroup of . We say that is **potentially characteristic** with respect to , or conditional to , if there exists a group containing such that is a characteristic subgroup of .

## Particular cases

Group property | Property of being a potentially characteristic subgroup |
---|---|

any group | normal subgroup (proof: NPC theorem) |

finite group | normal subgroup of finite group (proof: Finite NPC theorem) |

group of prime power order | finite-p-potentially characteristic subgroup |

group with fixed set of prime divisors | finite-pi-potentially characteristic subgroup |