# CN-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

*This group property is obtained by applying the condition that the centralizer of every element satisfies the following group property:* Abelian group

## Contents

## History

### Origin

The concept and terminology of **CN-groups** was introduced by Michio Suzuki in his attempts to solve Burnside's conjecture (which later became the odd-order theorem).

## Definition

Below are listed some **equivalent definitions** of CN-group, also called **centralizer is nilpotent** group:

No. | Shorthand | A group is termed a CN-group if ... | A group is termed a CN-group if ... |
---|---|---|---|

1 | element centralizers are nilpotent | the centralizer of any non-identity element is a nilpotent subgroup. | for every non-identity element , the centralizer (i.e., the set of all elements of that commute with ) is a nilpotent subgroup of . |

2 | subgroup centralizers are nilpotent | the centralizer of any nontrivial subgroup is a nilpotent subgroup. | for every nontrivial subgroup of , the centralizer is a nilpotent subgroup of . |

3 | subgroups: nilpotent or centerless | every subgroup of the group is either a nilpotent group or a centerless group. | for every nontrivial subgroup of , either is nilpotent or is centerless, i.e., the center of is trivial. |

### Equivalence of definitions

`Further information: equivalence of definitions of CN-group`

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | Yes | direct from version (3) of the definition | Suppose is a CN-group and is a subgroup of . Then, is also a CN-group. |

finite direct product-closed group property | Yes | Suppose and are CN-groups. Then, the external direct product is also a CN-group. |

## Facts

- For finite CN-groups: Sylow subgroups for distinct primes in CN-group centralize each other iff they have non-identity elements that centralize each other
- Quotient group of finite CN-group by solvable normal subgroup is CN-group

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

abelian group | the whole group is abelian | (via nilpotent) | CA-group|FULL LIST, MORE INFO | |

nilpotent group | the whole group is nilpotent | (obvious) | CN not implies nilpotent | |FULL LIST, MORE INFO |

CA-group | the centralizer of every non-identity element is abelian | follows from abelian implies nilpotent | any nilpotent non-abelian group, use a central element | |FULL LIST, MORE INFO |