Properties of groups

CA-group

In mathematics, in the realm of group theory, a group is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in the Feit–Thompson theorem and the classification of finite simple groups. Several important infinite groups are CA-groups, such as free groups, Tarski monsters, and some Burnside groups, and the locally finite CA-groups have been classified explicitly. CA-groups are also called commutative-transitive groups (or CT-groups for short) because commutativity is a transitive relation amongst the non-identity elements of a group if and only if the group is a CA-group. (Wikipedia).

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Related pages

Order (group theory) | Finite field | Free group | Group (mathematics) | CN-group | Frobenius group | Alternating group | Simple group | Finite group | Transitive relation | Brauer–Suzuki–Wall theorem | Classification of finite simple groups | Dihedral group | Locally finite group | Mathematics | Semidirect product | Group theory | Feit–Thompson theorem | Subgroup | Solvable group | Abelian group | Center (group theory)