Gruenberg group

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

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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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This is a variation of nilpotence|Find other variations of nilpotence | Read a survey article on varying nilpotence

Definition

A group $G$is said to be a Gruenberg group if it satisfies the following equivalent conditions:

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

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
nilpotent group				Baer group, Group in which every subgroup is subnormal, Group satisfying normalizer condition|FULL LIST, MORE INFO
group satisfying normalizer condition	no proper self-normalizing subgroup; or equivalently, every subgroup is ascendant			|FULL LIST, MORE INFO
Baer group	every cyclic subgroup is a subnormal subgroup			|FULL LIST, MORE INFO
group in which every subgroup is subnormal	every subgroup is a subnormal subgroup			Baer group, Group satisfying normalizer condition|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
locally nilpotent group	every finitely generated subgroup is nilpotent			|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