## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup $H$ of a group $G$ is termed a monadic second-order characteristic subgroup if there is no other subgroup $K$ of $G$ such that the monadic second-order theories of the group-subgroup pairs $(G,H)$ and $(G,K)$ coincide. In other words, $H$ can be distinguished from any other subgroup of $G$ using monadic second-order logic in the pure theory of the group $G$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
elementarily characteristic subgroup no other elementarily equivalent subgroup
monadic second-order purely definable subgroup can be defined using the pure theory of the group in monadic second-order language
purely definable subgroup can be defined purely using the first-order pure theory of the group

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Second-order characteristic subgroup
Characteristic subgroup