# Baer group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## Definition

A group $G$ is termed a Baer group if it satisfies the following equivalent conditions:

1. Every cyclic subgroup of $G$ is a subnormal subgroup of $G$.
2. Every finitely generated subgroup of $G$ is subnormal in $G$.
3. Every finitely generated subgroup of $G$ is a nilpotent subnormal subgroup: it is nilpotent as a group and subnormal as a subgroup.

For a finite group, this is equivalent to being a finite nilpotent group.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions