Surjective endomorphism-balanced subgroup

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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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If the ambient group is a finite group, this property is equivalent to the property: characteristic subgroup
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This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity

Definition

Symbol-free definition

A subgroup of a group is termed surjective endomorphism-balanced if every surjective endomorphism of the whole group restricts to a surjective endomorphism of the subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed surjective endomorphism-balanced if for any surjective endomorphism $\sigma$ of $G$, $\sigma (H)=H$.

Formalisms

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Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
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In the function restriction formalism, the property of being highly strictly characteristic is expressed as:

Surjective endomorphism $\to$ surjective endomorphism

Thus, it is a balanced subgroup property (function restriction formalism).

Relation with other properties

Stronger properties

• Completely strictly characteristic subgroup

Weaker properties

• Strictly characteristic subgroup (also called distinguished subgroup)
• Right-transitively strictly characteristic subgroup
• Characteristic subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Transitivity follows directly from the fact that the property is a balanced subgroup property.

Further information: Balanced implies transitive

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).
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