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Retraction-invariant subgroup

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) \le 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

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

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
View other monadic second-order subgroup properties

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.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
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

Conjunction with other properties

Weaker properties

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