# Endomorphism kernel

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 $H$ of a group $G$ is termed an endomorphism kernel in $G$ 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. $H$ is a normal subgroup of $G$ and there is a subgroup $M$ of $G$ such that the quotient group $G/H$ is isomorphic to $M$.
2 endomorphism kernel there is an endomorphism of the whole group whose kernel is precisely the subgroup. there is an endomorphism $\sigma$ of $G$ such that the kernel of $\sigma$ is $H$

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
transitive subgroup property No endomorphism kernel is not transitive It is possible to have group $H \le K \le G$ such that $H$ is an endomorphism kernel in $K$ and $K$ is an endomorphism kernel in $G$ but $H$ is not an endomorphism kernel in $G$.
intermediate subgroup condition No endomorphism kernel does not satisfy intermediate subgroup condition It is possible to have groups $H \le K \le G$ such that $H$ is an endomorphism kernel in $G$ but is not an endomorphism kernel in $K$.
quotient-transitive subgroup property Yes endomorphism kernel is quotient-transitive Suppose $H \le K \le G$ are groups such that $H$ is an endomorphism kernel in $G$ and $K/H$ is an endomorphism kernel in $G/H$. Then, $K$ is an endomorphism kernel in $G$.
trim subgroup property Yes obvious in any group $G$, the trivial subgroup and the whole group $G$ 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
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)