Gruenberg group: Difference between revisions
No edit summary |
|||
| (7 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
{{semistddef}} | |||
{{group property}} | {{group property}} | ||
{{variationof|nilpotence}} | {{variationof|nilpotence}} | ||
==Definition== | ==Definition== | ||
A [[group]] <math>G</math>is said to be a '''Gruenberg group''' if it satisfies the following equivalent conditions: | |||
# Every [[cyclic group|cyclic]] subgroup of <math>G</math> is [[ascendant subgroup|ascendant]] in <math>G</math>. | |||
# Every [[finitely generated group|finitely generated]] subgroup of <math>G</math> is [[ascendant subgroup|ascendant]] in <math>G</math>. | |||
# Every [[finitely generated group|finitely generated]] subgroup of <math>G</math> is [[ascendant subgroup|ascendant]] in <math>G</math> and [[nilpotent group|nilpotent as a group]]. | |||
==Relation with other properties== | ==Relation with other properties== | ||
| Line 13: | Line 15: | ||
===Stronger properties=== | ===Stronger properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::nilpotent group]] || || || || {{intermediate notions short|Gruenberg group|nilpotent group}} | |||
|- | |||
| [[Weaker than::group satisfying normalizer condition]] || no proper self-normalizing subgroup; or equivalently, ''every'' subgroup is ascendant || || || {{intermediate notions short|Gruenberg group|group satisfying normalizer condition}} | |||
|- | |||
| [[Weaker than::Baer group]] || every cyclic subgroup is a [[subnormal subgroup]] || || || {{intermediate notions short|Gruenberg group|Baer group}} | |||
|- | |||
| [[Weaker than::group in which every subgroup is subnormal]] || every subgroup is a [[subnormal subgroup]] || || || {{intermediate notions short|Gruenberg group|group in which every subgroup is subnormal}} | |||
|} | |||
===Weaker properties=== | ===Weaker properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::locally nilpotent group]] || every finitely generated subgroup is nilpotent || || || {{intermediate notions short|locally nilpotent group|Gruenberg group}} | |||
|} | |||
== References == | |||
=== Textbook references === | |||
{| class="sortable" border="1" | |||
! Book !! Page number !! Chapter and section !! Contextual information !! View | |||
|- | |||
| {{booklink-defined-tabular|RobinsonGT|353|Section 12.2|definition introduced in paragraph}} following 12.2.8 || [https://books.google.com/books?id=BFrTBwAAQBAJ&pg=PA348 Google Books] | |||
|} | |||
Latest revision as of 06:21, 17 April 2017
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 properties
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 | |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 |