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				|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			|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