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 non-identity 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 simple groups">simple group">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 because commutativity is a transitive relation amongst the non-identity elements of a group if and only if the group is a CA-group.
History
Locally finite CA-groups were classified by several mathematicians from 1925 to 1998. First, finite CA-groups were shown to be simple or solvable in. Then in the Brauer–Suzuki–Wall theorem, finite CA-groups of even order were shown to be Frobenius groups, abelian groups, or projective special linear groups over a finite field of even order, PSL for f ≥ 2. Finally, finite CA-groups of odd order were shown to be Frobenius groups or abelian groups in, and so in particular, are never nonabelian simple.CA-groups were important in the context of the classification of finite simple groups. Michio Suzuki showed that every finite, simple, nonabelian CA-group is of even order. This result was first extended to the Feit–Hall–Thompson theorem showing that finite, simple, nonabelian CN-groups had even order, and then to the Feit–Thompson theorem which states that every finite, simple, nonabelian group is of even order. A textbook exposition of the classification of finite CA-groups is given as example 1 and 2 in. A more detailed description of the Frobenius groups appearing is included in, where it is shown that a finite, solvable CA-group is a semidirect product of an abelian group and a fixed-point-free automorphism, and that conversely every such semidirect product is a finite, solvable CA-group. Wu also extended the classification of Suzuki et al. to locally finite groups.