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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DISJUNCTIVE DEFINITION: This article defines a term where the definition (or at least, one of the equivalent definitions) has the form "every A is a B or a C."

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 $G$ 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 $x\in G$, the centralizer $C_{G}(x)$ (i.e., the set of all elements of $G$ that commute with $x$) is an abelian subgroup of $G$.
2	subgroup centralizers are abelian	the centralizer of any nontrivial subgroup is an abelian subgroup.	for every nontrivial subgroup $H$ of $G$, the centralizer $C_{G}(H)$ is an abelian subgroup of $G$.
3	subgroups: abelian or centerless	every subgroup of the group is either an abelian group or a centerless group.	for every nontrivial subgroup $H$ of $G$, either $H$ is abelian or $H$ is centerless, i.e., the center of $H$ 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 $G$ is a CA-group and $H$ is a subgroup of $G$. Then, $H$ is also a CA-group.
direct product-closed group property	Yes		Suppose $G_{i},i\in I$ are CA-groups. Then, the external direct product $\prod _{i\in I}G_{i}$ 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