Subgroup property

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This article is about a general term. A list of important particular cases (instances) is available at Category: Subgroup properties

Definition

Symbol-free definition

A subgroup property is a map from the collection of all group-subgroup pairs to the two-element set (true, false), with the property that two equivalent subgroups (viz equivalent group-subgroup pairs) either both have the property or both do not have the property. The subgroup property space is the collection of all subgroup properties.

Definition with symbols

A subgroup property $p$ is defined as a map that takes as input a pair $(H,G)$ (where $H$ is a subgroup of $G$) and outputs either True or False in such a way that if $H_1$ ≤ $G_1$ and $H_2$ ≤ $G_2$ are equivalent as group-subgroup pairs then:

$(H_1,G_1)$ satisfies $p$ if and only if $(H_2,G_2)$ does.

We say loosely that $H$ satisfies $p$ in $G$ or that $H$ is a subgroup with property $p$ in $G$.

Caution

Whether a subgroup satisfies the property in the given group cannot be completely decided by looking at the isomorphism class of the group and the subgroup -- the manner in which the subgroup sits inside the group is also important.

Examples

Important examples of subgroup properties

Roughly speaking, the idea behind a subgroup property is that given any group, and any subgroup, the subgroup either satisfies the property inside the group or it does not satisfy. A natural example of a subgroup property is the property of being a normal subgroup.

Given any subgroup $H$ of a group $G$, we either have that $H$ is normal in $G$ or that $H$ is not normal in $G$.

Other important subgroup properties are such as the property of being a maximal subgroup, a maximal normal subgroup, a characteristic subgroup and so on.

Structure of the subgroup property space

Partial order

As for any property space, there is a natural partial order on the subgroup property space. Given subgroup properties $p$ and $q$, we say that $p$ is stronger than $q$, or $p$ is smaller than $q$, or $p$ implies $q$, if whenever a subgroup satisfies property $p$ in the whole group, it also satisfies property $q$ in the whole group.

The tautology, fallacy, trivial and improper properties

The two extreme elements of the subgroup property space with respect to the partial order are the tautology -- the property satisfied by all subgroups, and the fallacy -- the property satisfied by no automorphism.

Apart from these, there are two other important properties:

• The [[improper property: This is the property satisfied by any group as a subgroup of itself.
• The trivial property: This is the property satisfied by the trivial subgroup in any group.

A subgroup property that implies the improper property is termed identity-true, and a subgroup property that implies the trivial property is termed trivially true. A subgroup property that is both trivially true and identity-true is termed trim.

Conjunction of subgroup properties

Given a collection of subgroup properties, the conjunction of those subgroup properties is defined as the property that a subgroup satisfies if and only if it satisfies all the given subgroup properties. Thus, conjunction is the analogue of logical conjunction, and of set-theoretic intersection. Note, however, that conjunction is very different from the intersection operator on subgroup properties.

Disjunction of subgroup properties

Given a collection of subgroup properties, the disjunction of those subgroup properties is defined as the property that a subgroup satisfies if and only if it satisfies at least one of the given subgroup properties. Thus, disjunction is the analogue of logical disjunction, or of set-theoretic union. Note, however, that disjunction is very different from the join operator on subgroup properties.

Operators on the subgroup property space

A full list is available at Category: Subgroup property operators.

The composition operator

Further information: composition operator

The composition operator on the subgroup property space is an associative binary operation from the subgroup property space to itself, which sends a pair of properties $p$ and $q$ to the property $\alpha$ such that:

A subgroup $H$ satisfies property $\alpha$ in a group $G$ if there is a subgroup $K$ between $H$ and $G$ such that $H$ satisfies $p$ in $K$ and $K$ satisfies $q$ as a subgroup of $G$.

The identity element for the composition operator is the identity property. The nil element for the composition operator is the fallacy. Since the composition operator is associative, it admits notions of transiters with the transiter master theorems holding.

The upper and lower hook operators

The upper and lower hook operators arise from the same idea of composition but with different inputs and different outputs.

Particular cases of these operators are:

• The potentially operator
• The intermediately operator
• The left hereditarily operator
• The right hereditarily operator
• The upward-closed operator

The intersection operator

Further information: intersection operator The intersection operator sends a pair of properties $p$ and $q$ to the property $\alpha$ such that:

A subgroup $H$ of a group $G$ satisfies $\alpha$ in $G$ if and only if there exist subgroups $H_1$ and $H_2$ of $G$ such that $H = H_1$ ∩ $H_2$, $H_1$ satisfies $p$ and $H_2$ satisfies $q$.

The join operator

Further information: join operator

The join operator sends a pair of properties $p$ and $q$ to the property $\alpha$ such that:

A subgroup $H$ of a group $G$ satisfies $\alpha$ in $G$ if and only if there exist subgroups $H_1$ and $H_2$ of $G$ such that $H_1$ satisfies $p$, $H_2$ satisfies $q$, and $H$ is the subgroup generated by $H_1$ and $H_2$.

Formalisms for subgroup properties

The function restriction formalism

Further information: function restriction formalism

The function restriction formalism expresses subgroup properties that can be described in terms of the way they relate functions from the group to itself with functions from the subgroup to itself. More precisely, let $a$ and $b$ be two function properties. Then:

$a$ → $b$

is the subgroup property defined as follows: every function from the group to itself satisfying $a$ restricts to a function from the subgroup to itself satisfying $b$.

Such an expression is termed a function restriction formal expression. Function restriction formal expressions are available for properties such as being normal, characteristic, and being a central factor. These formal expressions allow various manipulations.

The relation implication formalism

Further information: relation implication operator

The relation implication formalism (or relation implication operator) takes as input two subgroup relations $a$ and $b$ and outputs the property of being a subgroup such that any subgroup to which it is related via $a$ is also such that they are related via $b$ (if the relation is not transitive, we have to be mindful of the order).