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 endomorphic 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 endomorphic 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

• Complemented normal subgroup
• Direct factor

Weaker properties

• Normal subgroup

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.