Invariance property

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BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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This article is about a general term. A list of important particular cases (instances) is available at Category:Invariance properties

History

The term invariant subgroup was first used for normal subgroups, because normality can be characterized as the invariance property with respect to inner automorphisms. Later, when Frobenius considered the concept of characteristic subgroup, he wanted to name it invariant subgroup but refrained from doing so because the term was in vogue for normal subgroup. Other notions of strictly invariant subgroup and fully invariant subgroup were also considered.

Although in the precise sense in which we are using here, the term invariance property is nonstandard, it has been used in similar sense in various standard texts.

Definition

Symbol-free definition

A subgroup property is termed an invariance property if, for any group, there is a collection of functions from the group to itself such that a subgroup of the group satisfies property if and only if it is invariant under all the functions in the collection.

Definition with symbols

A subgroup property is termed an invariance property if, for any group , there is a collection of functions such that a subgroup of satisfies if and only if is invariant under all functions in .

In terms of the invariant subgroup operator

An invariance property is precisely a property that is obtained by applying the invariant subgroup operator to a function property.

In terms of the function restriction formalism

The invariance property with respect to a function property is, in the function restriction formalism, expressible as:

→ function

In other words, every function on the whole group satisfying property restricts to a function on the subgroup.

Important invariance properties

Normality

Normality is an invariance property, obtained by applying the invariant subgroup operator to the function property of being an inner automorphism. In other words, a subgroup is normal if and only if every inner automorphism on the whole group takes the subgroup to within itself.

Characteristicity

Characteristicity is an invariance property, obtained by applying the invariant subgroup operator to the function property of being a automorphism. In other words, a subgroup is characteristic if and only if every automorphism of the whole group takes the subgroup to within itself.

All subgroup-defining functions yield characteristic subgroups. Thus, the center, commutator, Frattini subgroup, perfect core, hypercenter of a group are all characteristic.

Strict characteristicity

Strict characteristicity is an invariance property, obtained by applying the invariant subgroup operator to the function property of being a surjective endomorphism. In other words, a subgroup is strictly characteristic if and only if every surjective endomorphism of the whole group takes the subgroup to within itself.

The center of a group is always strictly characteristic. In fact, every term in the upper central series is strictly characteristic. More generally, any bound-word subgroup is strictly characteristic.

Full invariance

Full invariance is an invariance property, obtained by applying the invariant subgroup operator to the function property of being an endomorphism. In other words, a subgroup is fully invariant (also called fully characteristic) if and only if every automorphism of the whole group takes the subgroup to within itself.

Every verbal subgroup is fully invariant, and thus, in particular, the commutator subgroup as well as the members of the derived series and the lower central series, are all fully invariant. More generally, any existentially bound-word subgroup is fully invariant.

Relation with other metaproperties

Stronger metaproperties

Weaker metaproperties

Weaker metaproperties (subject to further conditions)

Transitivity

A subgroup property is said to be transitive if whenever has the property as a subgroup of and has the property as a subgroup of , then also has the property as a subgroup of . To determine whether an invariance property is transitive, we can use the technique of right tightening. The following turn out to be true:

  • The property of being normal is not transitive. In fact, its left transiter is the property of being characteristic.
  • The property of being characteristic is transitive.
  • The property of being strictly characteristic is not transitive.
  • The property of being fully invariant is transitive.