1-automorphism-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

Definition with symbols

A subgroup H of a group G is termed a 1-automorphism-invariant subgroup if any 1-automorphism of G sends H to itself. In other words, for every 1-automorphism \varphi of G, \varphi(H) \subseteq H.

Formalisms

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

The property of being a 1-automorphism-invariant subgroup can be expressed as an invariance property:

1-automorphism \to Function

In other words, H is a 1-automorphism-invariant subgroup of a group G if and only if every 1-automorphism of G restricts to a function from H to itself.

Alternative function restriction expressions are:

Examples

  • In any group, the whole group and the trivial subgroup are 1-automorphism-invariant.
  • In a cyclic group, every subgroup is 1-automorphism-invariant.
  • If the set of elements of order n form a subgroup for any particular n, then that subgroup is 1-automorphism-invariant. This is because 1-automorphisms preserve the orders of elements.

Relation with other properties

Stronger properties

Weaker properties

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

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

Join-closedness

This subgroup property is not join-closed, viz., it is not true that a join of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not join-closed