CA-group

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

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

Symbol-free definition

A group is termed a CA-group or a Centralizer is Abelian group if it satisfies the following conditions:

The centralizer of any nontrivial element is an Abelian subgroup.
The centralizer of any nontrivial subgroup is an Abelian subgroup.

Definition with symbols

A group $G$ is termed a CA-group or a Centralizer is Abelian group if it satisfies the following conditions:

For any $e\neq x\in G$ , the group $C_{G}(x)$ is Abelian
For any nontrivial subgroup $H\leq G$ , the group $C_{G}(H)$ is Abelian

Relation with other properties

Stronger properties

Abelian group

Weaker properties

CN-group

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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If $G$ is a CA-group and $H\leq G$ is a subgroup, then for any $x\in H$ , $C_{H}(x)=H\cap C_{G}(x)$ . Hence, if $C_{G}(x)$ is Abelian, so is $C_{H}(x)$ . Thus, any subgroup of a CA-group is a CA-group.

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
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If $G_{1}$ and $G_{2}$ are CA-groups, so is $G_{1}\times G_{2}$ , because the centralizer of the element $(x_{1},x_{2})\in G_{1}\times G_{2}$ is $C_{G_{1}}(x_{1})\times C_{G_{2}}(x_{2})$ .