Invariance property

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.

Symbol-free definition
A subgroup property $$p$$ 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 $$p$$ if and only if it is invariant under all the functions in the collection.

Definition with symbols
A subgroup property $$p$$ is termed an invariance property if, for any group $$G$$, there is a collection $$F$$ of functions such that a subgroup $$H$$ of $$G$$ satisfies $$p$$ if and only if $$H$$ is invariant under all functions $$f$$ in $$F$$.

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 $$q$$ is, in the function restriction formalism, expressible as:

$$p$$ &rarr; function

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

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.

Stronger metaproperties

 * Endo-invariance property: This is the invariance property with respect to a function property that is satisfied only by endomorphisms.
 * Auto-invariance property: This is the invariance property with respect to a group-closed automorphism property.

Weaker metaproperties

 * Multi-invariance property
 * Strongly intersection-closed subgroup property:
 * Intersection-closed subgroup property
 * Identity-true subgroup property
 * ACU-closed subgroup property

Weaker metaproperties (subject to further conditions)

 * Trivially true subgroup property: If every function satisfying $$p$$ fixes the identity element, then the trivial subgroup satisfies property $$\alpha$$.
 * Join-closed subgroup property: If every function satisfying $$p$$ is an endomorphism (that is, we are in the endo-invariance property case), then the subgroup generated by any family of subgroups each with property $$\alpha$$ also has property $$\alpha$$.
 * Transfer condition: An invariance property with respect to an extensibility-stable function property must satisfy the transfer condition: namely if $$H$$ satisfies the invariance property in $$G$$, and $$K$$ is a subgroup of $$G$$, then $$H$$ &cap; $$K$$ satisfies the property in $$K$$.

Transitivity
A subgroup property is said to be transitive if whenever $$G$$ has the property as a subgroup of $$H$$ and $$H$$ has the property as a subgroup of $$K$$, then $$G$$ also has the property as a subgroup of $$K$$. 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.