First-order subgroup property

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

Definition

Symbol-free definition

A subgroup property is said to be a first-order subgroup property if it can be expressed using a first-order formula, viz a formula that allows:

• Logical operations (conjunction, disjunction, negation, and conditionals)
• Equality testing
• Quantification over elements of the group and subgroup (this in particular allows one to test membership of an element of the group, in the subgroup)
• Group operations (multiplication, inversion and the identity element)

Things that are not allowed are quantification over other subgroups, quantification over automorphisms, and quantification over supergroups.

Importance

First-order language is severely constricted, at least when it comes to subgroup properties. Hence, not only are there very few first-order subgroup properties of interest, also, very few of the subgroup property operators preserve the first-order nature.

Examples

Normality

Normality is a first-order subgroup property as can be seen from the following definition: a subgroup $N$ of a group $G$ is termed normal if the following holds:

$\mathrm{\forall }g\in G,h\in N,gh{g}^{-1}\in N$

The formula is universal of quantifier rank 1.

Centrality

A subgroup is a central subgroup if it lies inside the center, or equivalently, if every element in the subgroup commutes with every element in the group.

Clearly, the property of being a central subgroup is first-order.

The formula is universal of quantifier rank 1.

Central factor

A subgroup is a central factor if every element in the group can be expressed as a product of an element in the subgroup and an element in the centralizer. This can naturally be expressed as a first-order formula of quantifier rank 3 with the outermost layer being universal.

$\mathrm{\forall }g\in G,\mathrm{\exists }h\in H.\mathrm{\forall }x\in H,hx{h}^{-1}=gx{g}^{-1}$

Relation with formalisms

Function restriction formalism

The general question of interest: given a subgroup property with a function restriction expression $a\to b$, can we use the expression to give a first-order definition for the subgroup property? It turns out that the following suffice:

• $a$ should be a first-order enumerable function property (this condition is much stronger than just being a first-order function property because we are not allowed to directly quantify over functions.
• $b$ should be a first-order function property in the sense that given any function, it must be possible to give a first-order formula that outputs whether or not the function satisfies $b$.

The primary example of a first-order enumerable function property is the property of being an inner automorphism. Most function properties that we commonly enoucnter are first-order (that is, they can be tested/verified using first-order formulae).