Baer group

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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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Definition

A group G is termed a Baer group if it satisfies the following equivalent conditions:

  1. Every cyclic subgroup of G is a subnormal subgroup of G.
  2. Every finitely generated subgroup of G is subnormal in G.
  3. Every finitely generated subgroup of G is a nilpotent subnormal subgroup: it is nilpotent as a group and subnormal as a subgroup.

For a finite group, this is equivalent to being a finite nilpotent 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 in which every subgroup is subnormal |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Gruenberg group every cyclic subgroup is ascendant |FULL LIST, MORE INFO
locally nilpotent group every finitely generated subgroup is nilpotent (via Gruenberg group) (via Gruenberg group) Gruenberg group|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