# Elementarily characteristic subgroup

From Groupprops

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]

This term is related to: model theory

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BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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 | Purely definably generated subgroup|FULL LIST, MORE INFO | |||

verbal subgroup of finite type | Purely definable subgroup|FULL LIST, MORE INFO | |||

marginal subgroup of finite type | Purely definable subgroup|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 | Purely definable subgroup, Purely definably generated subgroup|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 | Monadic second-order characteristic subgroup|FULL LIST, MORE INFO | |

monadic second-order characteristic subgroup | |FULL LIST, MORE INFO |