# 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]

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Equivalent definitions in tabular format

No. | Shorthand | A subgroup of a group is termed an endomorphism kernel if ... | A subgroup of a group is termed an endomorphism kernel in if ... |
---|---|---|---|

1 | normal, quotient isomorphic to subgroup | it is normal in the whole group and its quotient group is isomorphic to some subgroup of the whole group. | is a normal subgroup of and there is a subgroup of such that the quotient group is isomorphic to . |

2 | endomorphism kernel | there is an endomorphism of the whole group whose kernel is precisely the subgroup. | there is an endomorphism of such that the kernel of is |

## Metaproperties

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

transitive subgroup property | No | endomorphism kernel is not transitive | It is possible to have group such that is an endomorphism kernel in and is an endomorphism kernel in but is not an endomorphism kernel in . |

intermediate subgroup condition | No | endomorphism kernel does not satisfy intermediate subgroup condition | It is possible to have groups such that is an endomorphism kernel in but is not an endomorphism kernel in . |

quotient-transitive subgroup property | Yes | endomorphism kernel is quotient-transitive | Suppose are groups such that is an endomorphism kernel in and is an endomorphism kernel in . Then, is an endomorphism kernel in . |

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

## Relation with other properties

### Stronger properties

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

complemented normal subgroup | complemented normal implies endomorphism kernel | endomorphism kernel not implies complemented normal | Intermediately endomorphism kernel|FULL LIST, MORE INFO | |

direct factor | Complemented normal subgroup, Intermediately endomorphism kernel|FULL LIST, MORE INFO | |||

subgroup of finite abelian group | follows from subgroup lattice and quotient lattice of finite abelian group are isomorphic | (trivial subgroup, whole group are endomorphism kernels even in non-abelian groups) | Intermediately endomorphism kernel|FULL LIST, MORE INFO |

### Weaker properties

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

quotient-powering-invariant subgroup | endomorphism kernel implies quotient-powering-invariant | any normal subgroup of a finite group that is not an endomorphism kernel works. | |FULL LIST, MORE INFO | |

powering-invariant normal subgroup | (via quotient-powering-invariant) | (via quotient-powering-invariant) | Quotient-powering-invariant subgroup|FULL LIST, MORE INFO | |

powering-invariant subgroup | (via quotient-powering-invariant) | (via quotient-powering-invariant) | Powering-invariant normal subgroup, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO | |

normal subgroup | (by definition) | normal not implies endomorphism kernel | Powering-invariant normal subgroup, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO |

## Effect of property operators

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Operator | Meaning | Result of application | Proof |
---|---|---|---|

potentially operator | endomorphism kernel in some larger group | normal subgroup | normal implies potential endomorphism kernel |

intermediately operator | endomorphism kernel in every intermediate subgroup | intermediately endomorphism kernel | (by definition) |