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]
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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VIEW 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
Symbol-free definition
A group is said to be a Gruenberg group if every cyclic subgroup of it is ascendant.
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 | ||
| 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 |