Invariance property

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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 $p$ 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 from $G$ to itself 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:

$q\to$ function

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

Equivalence of definitions

While for the most part the equivalence of definitions is straightforward, there is a nuance regarding the equivalence between considering arbitrary collections of functions for each individual group and thinking of a function property. More specifically, even if for each group $G$ there is a collection of functions $F$ (dependent on $G$) how do we know that we can come up with a function property: something definable universally and invariant under isomorphisms?

To establish this, we can use one of many potential closure operations to go from the arbitrary collection of functions to the most complete collection of functions possible. The most expansive one is to define the function property in terms of the subgroup property: it's the property of being a function from $G$ to itself that preserves all subgroups satisfying property $p$. Since $p$ is a subgroup property, that automatically imparts a "property" nature (and in particular preservation under isomorphisms) to the induced function property. The function property that we obtain this way is left tight: it's the most generous possible function property we can come up with for which $p$ is an invariance property. In practice, this might be too unconstrained a function property, but it gives proof of existence.

In many cases, such as the case of an auto-invariance property or endo-invariance property, we can limit the functions to automorphisms or endomorphisms respectively.

The narrowest closure operation we can do is to consider the union of the orbits under all automorphisms of $G$ of the functions. As long as our selection is consistent across all pairwise isomorphic groups, this works. However, if the collection of functions cannot be described in language, then this is not a satisfying way of formulating a "function property".

Examples

Subgroup property	Function property for which it is the invariance property	Function restriction expression	Further comments
normal subgroup	inner automorphism	inner automorphism $\to$ function	Normality is an auto-invariance property since being an inner automorphism is a group-closed automorphism property. In other words, we can write it as inner automorphism $\to$ automorphism. This is a right tight function restriction expression (i.e., we cannot make the right side stronger than automorphism); see inner automorphism to automorphism is right tight for normality.
characteristic subgroup	automorphism	automorphism $\to$ function	Characteristicity is an auto-invariance property since being an automorphism is a group-closed automorphism property. In other words, we can write it as automorphism $\to$ automorphism. This also shows that it is a balanced subgroup property.
fully invariant subgroup	endomorphism	endomorphism $\to$ function	Full invariance is an endo-invariance property. In other words, we can write it as endomorphism $\to$ endomorphism. This also shows that it is a balanced subgroup property.
powering-invariant subgroup	rational power map	rational power map $\to$ function	It can also be written as rational power map $\to$ rational power map, and hence is a balanced subgroup property.
local powering-invariant subgroup	local powering	local powering $\to$ function	It can also be written as local powering $\to$ local powering, and hence is a balanced subgroup property.

Relation with other metaproperties

Stronger metaproperties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
endo-invariance property	invariance property with respect to a function property that is satisfied only by endomorphisms	(by definition)
auto-invariance property	invariance property with respect to a group-closed automorphism property	(by definition)		|FULL LIST, MORE INFO

Weaker metaproperties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
multi-invariance property	invariance property with respect to a collection of functions of potentially varying arities	set all functions to have arity 1
strongly intersection-closed subgroup property	closed under arbitrary intersections including the empty intersection (whole group)	invariance implies strongly intersection-closed	strongly intersection-closed not implies invariance
intersection-closed subgroup property	closed under arbitrary intersections but not including the empty intersection	(via strongly intersection-closed)	(via strongly intersection-closed)
identity-true subgroup property	true for any group as a subgroup of itself	(by definition)	any identity-true property that is not intersection-closed will suffice; for instance, the property of having odd index, or having index at most 2
union-closed subgroup property	closed under union, if the union also happens to be a subgroup	similar proof as for intersection-closed
ACU-closed subgroup property	closed under unions of ascending chains of subgroups	(via union-closed)		|FULL LIST, MORE INFO

Weaker metaproperties (subject to further conditions)

Below, we denote by $q$ the function property and by $p$ the invariance property.

• Trivially true subgroup property: If every function satisfying $q$ fixes the identity element, then the trivial subgroup satisfies property $p$.
• Join-closed subgroup property: If every function satisfying $q$ 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 $p$ also has property $p$. For full proof, refer: Endo-invariance implies join-closed
• 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 $H$ ∩ $K$ 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.