Retraction-invariant 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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Definition

Symbol-free definition

A subgroup of a group is termed retraction-invariant if any retraction (viz, idempotent endomorphism) of the whole group takes the subgroup to within itself.

Definition with symbols

A subgroup $H$ of a group $G$ is termed retraction-invariant if, given any retraction (viz, idempotent endomorphism) $\sigma$ of $G$, $\sigma (H)\leq H$.

A retraction is an idempotent endomorphism, viz $\sigma$ is a retraction if and only if $\sigma (\sigma (g))=\sigma (g)$ for all $g$ in $G$.

Formalisms

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Monadic second-order description

This subgroup property is a monadic second-order subgroup property, viz., it has a monadic second-order description in the theory of groups
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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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Function restriction expression	$H$ is a retraction-invariant subgroup of $G$ if ...	This means that retraction-invariance is ...	Additional comments
retraction $\to$ function	every retraction of $G$ sends every element of $H$ to within $H$	the invariance property for retractions
retraction $\to$ endomorphism	every retraction of $G$ restricts to an endomorphism of $H$	the endo-invariance property for retractions; i.e., it is the invariance property for retraction, which is a property stronger than the property of being an endomorphism
retraction $\to$ retraction	every retraction of $G$ restricts to a retraction of $H$	the balanced subgroup property for retractions	Hence, it is a t.i. subgroup property, both transitive and identity-true

Relation with other properties

Stronger properties

• Fully invariant subgroup

Conjunction with other properties

• Retraction-invariant retract is the conjunction with the property of being a retract.
• Retraction-invariant direct factor is the conjunction with the property of being a direct factor.
• Retraction-invariant central factor is the conjunction with the property of being a central factor.
• Retraction-invariant normal subgroup is the conjunction with the property of being a normal subgroup.

Weaker properties

• Direct projection-invariant subgroup
• Retract-transfering subgroup: In other words, if $H$ is a retraction-invariant subgroup of $G$ and $K$ is a retract of $G$, then $H\cap K$ is a retract of $H$.

Incomparable with normality

Note that there are retraction-invariant subgroups which are not normal. In fact, in a simple group, every subgroup is retraction-invariant, although none except the trivial subgroup and the whole group are normal.

Further, we have examples of normal subgroups that are not retraction-invariant. For instance, the copy of $G$ in $G\times G$ is not invariant under the retraction $(g,h)\mapsto (g,g)$.

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.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
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

Any retraction-invariant subgroup of a retraction-invariant subgroup is retraction-invariant. This easily follows on account of retraction-invariance being a balanced subgroup property, that is, from the fact that its restriction formal expression has the same thing on the left side and on the right side.

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 whole group and the trivial subgroup are clearly retraction-invariant.

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

An arbitrary intersection of retraction-invariant subgroups is retraction-invariant. This follows from its being an invariance property. For full proof, refer: Invariance implies intersection-closed

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

An arbitrary join of retraction-invariant subgroups is retraction-invariant. This follows from its being an endo-invariance property. For full proof, refer: Endo-invariance implies join-closed