Elementarily characteristic subgroup

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]

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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

This term is related to: model theory
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This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity

Definition

Symbol-free definition

A subgroup of a group is said to be elementarily characteristic or first-order characteristic if there is no subgroup that is elementarily equivalently embedded.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
purely definable subgroup				|FULL LIST, MORE INFO
verbal subgroup of finite type				|FULL LIST, MORE INFO
marginal subgroup of finite type				|FULL LIST, MORE INFO
purely definably generated subgroup	has a generating set that is definable in the first-order theory of the group			|FULL LIST, MORE INFO
characteristic subgroup of finite group				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic subgroup	invariant under all automorphisms		characteristic not implies elementarily characteristic	|FULL LIST, MORE INFO
monadic second-order characteristic subgroup				|FULL LIST, MORE INFO