# Gruenberg group

From Groupprops

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]

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

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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

## Definition

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

- Every cyclic subgroup of is ascendant in .
- Every finitely generated subgroup of is ascendant in .
- Every finitely generated subgroup of is ascendant in 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 0387944613^{More info} |
353 | Section 12.2 | definition introduced in paragraph following 12.2.8 | Google Books |