Open main menu

Groupprops β

Auto-invariance property

Definition

Definition with symbols

A subgroup property p is termed an auto-invariance property if there is a group-closed automorphism property q such that for any subgroup H \le G, H has property p in G, iff any automorphism of G satisfying property p, sends H to within itself.

In terms of the function restriction formalism

A subgroup property p is termed an auto-invariance property if there is a group-closed automorphism property q, such that we can write the following function restriction expression for p:

p = q \to Function

In other words, a subgroup has subgroup property p if every automorphism with property q, for the whole group, restricts to a function from the subgroup to itself.

This is equivalent to the following function restriction expressions:

p = q \to Automorphism

and:

p = q \to Endomorphism

In other words, the restriction is automatically guaranteed to be an automorphism of the subgroup.

Equivalence of definitions

The equivalence of definitions follows from the elementary observation: restriction of automorphism to subgroup invariant under it and its inverse is automorphism.

Examples

Normal subgroups

Further information: normal subgroup

The property of normality is an auto-invariance property, where the group-closed automorphism property in question is the property of being an inner automorphism.

Characteristic subgroups

Further information: characteristic subgroup

The property of being characteristic is an auto-invariance property, where the group-closed automorphism property in question is the property of being any automorphism.

Relation with other metaproperties