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 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