Auto-invariance property: Difference between revisions

Latest revision as of 20:59, 7 October 2008

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
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VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

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 $H \leq G$ , $H$ 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

Weaker metaproperties

@@ Line 25: / Line 25: @@
 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==
@@ Line 44: / Line 48: @@
 ===Weaker metaproperties===
-* [[Endo-invariance property]]
+* [[Stronger than::Endo-invariance property]]
-* [[Invariance property]]
+* [[Stronger than::Invariance property]]
-* [[Strongly join-closed subgroup property]]
+* [[Stronger than::Strongly join-closed subgroup property]]
-* [[Strongly intersection-closed subgroup property]]
+* [[Stronger than::Strongly intersection-closed subgroup property]]
+* [[Stronger than::Automorphism-based relation-implication-expressible subgroup property]]
+* [[Stronger than::Normalizer-closed subgroup property]]
+* [[Stronger than::Centralizer-closed subgroup property]]
+* [[Stronger than::Commutator-closed subgroup property]]