# Monadic second-order subgroup property

From Groupprops

*This article defines a subgroup metaproperty in terms of the model-theoretic approach: viz by looking at the language in which the subgroup property can be defined*

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

## Contents

## Definition

### Symbol-free definition

A subgroup property is termed a **monadic second-order group property** if it can be expressed via monadic second-order logic; in other words, it can be expressed in a language which allows for:

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

Since we can quantify over all subsets, and can test closure under group operations, we can also quantify over all subgroups.