Socle is normality-preserving endomorphism-invariant

Statement

The socle of a group (defined as the join of all its minimal normal subgroups) is always a normality-preserving endomorphism-invariant subgroup, i.e., it contains its image under any normality-preserving endomorphism of the whole group. A normality-preserving endomorphism is an endomorphism that sends normal subgroups to normal subgroups.

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., socle) always satisfies a particular subgroup property (i.e., normality-preserving endomorphism-invariant subgroup)}
View subgroup property satisfactions for subgroup-defining functions $|$ View subgroup property dissatisfactions for subgroup-defining functions

Related facts

Applications

Property	Meaning	Proof that normality-preserving endomorphism-invariant implies this property	Proof that socle satisfies this property
strictly characteristic subgroup	invariant under surjective endomorphisms	normality-preserving endomorphism-invariant implies strictly characteristic	socle is strictly characteristic
direct projection-invariant subgroup	invariant under projections to direct factors	normality-preserving endomorphism-invariant implies direct projection-invariant	socle is direct projection-invariant
finite direct power-closed characteristic subgroup	finite direct power of subgroup is characteristic in corresponding finite direct power of whole group	normality-preserving endomorphism-invariant implies finite direct power-closed characteristic	socle is finite direct power-closed characteristic
characteristic subgroup	invariant under all automorphisms	(via strictly characteristic)	socle is characteristic