# Monadic second-order characteristic subgroup

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

## 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 |