Purely definably generated 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 purely definably generated if it has a generating set that is a definable subset in terms of the first-order theory of the group.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
purely definable subgroup the subgroup itself is definable
characteristic subgroup of finite group Purely definable subgroup|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
elementarily characteristic subgroup no other elementarily equivalently embedded subgroup Monadic second-order characteristic subgroup|FULL LIST, MORE INFO
characteristic subgroup invariant under every automorphism Elementarily characteristic subgroup|FULL LIST, MORE INFO