Monadic second-order characteristic subgroup

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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