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 ≥ 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.
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