# Baer group

From Groupprops

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 propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

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

- Every cyclic subgroup of is a subnormal subgroup of .
- Every finitely generated subgroup of is subnormal in .
- Every finitely generated subgroup of 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 0387944613^{More info} |
353 | Section 12.2 | definition introduced in paragraph following 12.2.8 | Google Books |