# Purely definably generated subgroup

From Groupprops

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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 |