Gruenberg group

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

Definition

A group Gis 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 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