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
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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
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 0387944613More info | 353 | Section 12.2 | definition introduced in paragraph following 12.2.8 | Google Books |