1-automorphism-invariant subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to 1-automorphism-invariant subgroup, all facts related to 1-automorphism-invariant subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity

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:

• 1-automorphism $\to$ 1-endomorphism
• 1-automorphism $\to$ 1-automorphism. This shows that the property of being a 1-automorphism-invariant subgroup is a balanced subgroup property (function restriction formalism) with respect to 1-automorphisms.

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

• 1-endomorphism-invariant subgroup

Weaker properties

• Quasiautomorphism-invariant subgroup
• Strong quasiautomorphism-invariant subgroup
• Characteristic subgroup
• Normal 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.
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