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