Center-fixing 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]

Definition

A subgroup of a group is termed a center-fixing automorphism-invariant subgroup if every center-fixing automorphism of the whole group sends the subgroup to within itself.

Metaproperties

Metaproperty Satisfied? Proof Statement with symbols
strongly intersection-closed subgroup property Yes Follows from invariance implies strongly intersection-closed If H_i, i \in I are all center-fixing automorphism-invariant subgroups of G, then so is the intersection of subgroups \bigcap_{i \in I} H_i.
strongly join-closed subgroup property Yes Follows from endo-invariance implies strongly join-closed If H_i, i \in I are all center-fixing automorphism-invariant subgroups of G, then so is the join of subgroups \langle H_i \rangle_{i \in I}.
trim subgroup property Yes direct from definition In any group G, both the whole group G and the trivial subgroup are center-fixing automorphism-invariant.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup invariant under all automorphisms characteristic implies center-fixing automorphism-invariant center-fixing automorphism-invariant not implies characteristic |FULL LIST, MORE INFO
central subgroup contained in the center central implies center-fixing automorphism-invariant center-fixing automorphism-invariant not implies central |FULL LIST, MORE INFO
center-fixing automorphism-balanced subgroup every center-fixing automorphism of the whole group restricts to a center-fixing automorphism of the subgroup. |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup invariant under all inner automorphisms |FULL LIST, MORE INFO

Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

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
Function restriction expression H is a center-fixing automorphism-invariant subgroup of G if ... This means that being center-fixing automorphism-invariant is ... Additional comments
center-fixing automorphism \to function every center-fixing automorphism of G sends every element of H to within H the invariance property for center-fixing automorphisms On account of this, it is a strongly intersection-closed subgroup property.
center-fixing automorphism \to endomorphism every center-fixing automorphism of G restricts to an endomorphism of H the endo-invariance property for center-fixing automorphisms; i.e., it is the invariance property for center-fixing automorphism, which is a property stronger than the property of being an endomorphism On account of this, it is a strongly join-closed subgroup property.
center-fixing automorphism \to automorphism every center-fixing automorphism of G restricts to an automorphism of H the auto-invariance property for center-fixing automorphisms; i.e., it is the invariance property for center-fixing automorphism, which is a group-closed property of automorphisms