# CA-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

DISJUNCTIVE DEFINITION: This article defines a term where the definition (or at least, one of the equivalent definitions) has the form "everyAis aBor aC."

## Contents

## History

### Origin

The concept and terminology of **CA-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 CA-group, also called **centralizer is abelian** group:

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

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

2 | subgroup centralizers are abelian | the centralizer of any nontrivial subgroup is an abelian subgroup. | for every nontrivial subgroup of , the centralizer is an abelian subgroup of . |

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

### Equivalence of definitions

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

## Metaproperties

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

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

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

## Relation with other properties

### Stronger properties

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

abelian group | (by definition) | CA not implies abelian | |FULL LIST, MORE INFO |

### Weaker properties

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

CN-group | |FULL LIST, MORE INFO |