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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
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions
Definition with symbols
A subgroup property is termed an auto-invariance property if there is a group-closed automorphism property such that for any subgroup , has property in , iff any automorphism of satisfying property , sends to within itself.
In terms of the function restriction formalism
In other words, a subgroup has subgroup property if every automorphism with property , for the whole group, restricts to a function from the subgroup to itself.
This is equivalent to the following function restriction expressions:
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.
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.
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
- Endo-invariance property
- Invariance property
- Strongly join-closed subgroup property
- Strongly intersection-closed subgroup property
- Automorphism-based relation-implication-expressible subgroup property
- Normalizer-closed subgroup property
- Centralizer-closed subgroup property
- Commutator-closed subgroup property