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]

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Definition

Symbol-free definition

A subgroup of a group is termed an endomorphism kernel if it satisfies the following equivalent conditions:

• It is normal and there is a subgroup of the group isomorphic to the quotient group
• There is an endomorphism of the group whose kernel is the given subgroup

Definition with symbols

A subgroup $H$ of a group $G$ is termed an endomorphism kernel if it satisfies the following conditions:

• There is a subgroup $K$ of $G$ such that $G/H\cong K$
• There is an endomorphism $\rho$ of $G$ such that the kernel of $\rho$ is $H$

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	|FULL LIST, MORE INFO
direct factor				|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)	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup		(by definition)	normal not implies endomorphism kernel

Metaproperties

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties

If $N$ is an endomorphic kernel in $G$, and $M$ is a subgroup containing $N$ such that $M/N$ is an endomorphic kernel in $G/N$.

The proof of this follows by simply composing the two endomorphisms.

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

The trivial subgroup is the kernel of the identity map, while the whole group is the kernel of the trivial endomorphism.