# Difference between revisions of "First-order subgroup property"

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{{subgroup metaproperty}} | {{subgroup metaproperty}} | ||

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+ | {{particularcases|[[Category:First-order subgroup properties]]}} | ||

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+ | ==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) | ||

+ | * Quantification over elements of the group and subgroup | ||

+ | * Group operations (multiplication, inversion and the identity element | ||

+ | |||

+ | Things that are ''not'' allowed are quantification over 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 <math>N</math> of a group <math>G</math> is termed '''normal''' if the following holds: | ||

+ | |||

+ | <math>\forall g \in G,h \in N, ghg^{-1} \in N</math> | ||

+ | |||

+ | 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. | ||

+ | |||

+ | <math>\forall g \in G (\exists h \in H, k \in G (\forall m \in H, km = mk))</math> | ||

+ | |||

+ | ==Relation with formalisms== | ||

+ | |||

+ | ===Function restriction formalism=== | ||

+ | |||

+ | The general question of interest: given a [[subgroup property]] with a [[function restriction formal expression]] <math>a \to b</math>, can we use the expression to give a first-order definition for the subgroup property? It turns out that the following suffice: | ||

+ | |||

+ | * <math>a</math> 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. | ||

+ | * <math>b</math> 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 <math>b</math>. | ||

+ | |||

+ | 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). |

## Revision as of 10:17, 27 February 2007

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

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

## 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)
- Quantification over elements of the group and subgroup
- Group operations (multiplication, inversion and the identity element

Things that are *not* allowed are quantification over 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 formal 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).