Baer group

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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
nilpotent group				|FULL LIST, MORE INFO
group in which every subgroup is subnormal				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Gruenberg group	every cyclic subgroup is ascendant			|FULL LIST, MORE INFO
locally nilpotent group	every finitely generated subgroup is nilpotent	(via Gruenberg group)	(via Gruenberg group)	|FULL LIST, MORE INFO

References

Textbook references

Book	Page number	Chapter and section	Contextual information	View
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info	353	Section 12.2	definition introduced in paragraph following 12.2.8	Google Books