# First-order subgroup property

Template:Subgroup metaproperty related to

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

## Contents

## 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 of a group is termed **normal** if the following holds:

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.

## Relation with formalisms

### Function restriction formalism

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

- 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.
- 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 .

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).