CA-group

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 Brauer–Suzuki–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, 2^f) for f ≥ 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 Feit–Hall–Thompson theorem showing that finite, simple, non-abelian, CN-groups had even order, and then to the Feit–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. 291–305). 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.

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 ≥ 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.