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    • Source: 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 Feitā€“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.


      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 (Weisner 1925). Then in the Brauerā€“Suzukiā€“Wall theorem (Brauer, Suzuki & Wall 1958), finite CA-groups of even order were shown to be Frobenius groups, abelian groups, or two dimensional projective special linear groups over a finite field of even order, PSL(2, 2f) for f ā‰„ 2. Finally, finite CA-groups of odd order were shown to be Frobenius groups or abelian groups in (Suzuki 1957), and so in particular, are never non-abelian simple.
      CA-groups were important in the context of the classification of finite simple groups. Michio Suzuki showed that every finite, simple, non-abelian, CA-group is of even order. This result was first extended to the Feitā€“Hallā€“Thompson theorem showing that finite, simple, non-abelian, CN-groups had even order, and then to the Feitā€“Thompson theorem which states that every finite, simple, non-abelian group is of even order. A textbook exposition of the classification of finite CA-groups is given as example 1 and 2 in (Suzuki 1986, pp. 291ā€“305). A more detailed description of the Frobenius groups appearing is included in (Wu 1998), 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.


      Examples


      Every abelian group is a CA-group, and a group with a non-trivial center is a CA-group if and only if it is abelian. The finite CA-groups are classified: the solvable ones are semidirect products of abelian groups by cyclic groups such that every non-trivial element acts fixed-point-freely and include groups such as the dihedral groups of order 4k+2, and the alternating group on 4 points of order 12, while the nonsolvable ones are all simple and are the 2-dimensional projective special linear groups PSL(2, 2n) for n ā‰„ 2. Infinite CA-groups include free groups, PSL(2, R), and Burnside groups of large prime exponent, (Lyndon & Schupp 2001, p. 10). Some more recent results in the infinite case are included in (Wu 1998), including a classification of locally finite CA-groups. Wu also observes that Tarski monsters are obvious examples of infinite simple CA-groups.


      Works cited

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