# Intermediately endomorphism kernel

From Groupprops

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]

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## Definition

Let be a group and be a subgroup of . We say that is **intermediately** (an) **endomorphism kernel** if for any intermediate subgroup (with , is an endomorphism kernel in .

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

transitive subgroup property | No | can use some of the same examples as for endomorphism kernel is not transitive | We can have groups such that is intermediately an endomorphism kernel in and is intermediately an endomorphism kernel in but is not intermediately an endomorphism kernel in . |

intermediate subgroup condition | Yes | obvious reasons | If are groups such that is intermediately an endomorphism kernel in , then is intermediately an endomorphism kernel in . |

quotient-transitive subgroup property | Yes | If are groups such that is intermediately an endomorphism kernel in and is intermediately an endomorphism kernel in , then is intermediately an endomorphism kernel in . | |

trim subgroup property | Yes | In any group, the whole group and the trivial subgroup are intermediately endomorphism kernels. |

## Relation with other properties

### Stronger properties

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

complemented normal subgroup | normal and has a permutable complement, i.e., part of an internal semidirect product | follows from complemented normal satisfies intermediate subgroup condition and complemented normal implies endomorphism kernel | |FULL LIST, MORE INFO | |

direct factor | (via complemented normal) | (via complemented normal) | |FULL LIST, MORE INFO | |

subgroup of finite abelian group | follows from its being an endomorphism kernel, which in turn follows from subgroup lattice and quotient lattice of finite abelian group are isomorphic | |FULL LIST, MORE INFO |

### Weaker properties

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

endomorphism kernel | kernel of an endomorphism | (by definition) | follows from endomorphism kernel does not satisfy intermediate subgroup condition | |FULL LIST, MORE INFO |

normal subgroup | (via endomorphism kernel) | (via endomorphism kernel) | |FULL LIST, MORE INFO |

## Formalisms

### In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: endomorphism kernel

View other properties obtained by applying the intermediately operator