The finite simple groups with at most two fusion classes of every order

Elements a, b of a group G are said to be fused if a = b(sigma) and to be inverse-fused if a = (b(-1))(sigma) for some sigma is an element of Aut(G). The fusion class of a is an element of G is the set {a(sigma) \ sigma is an element of Aut(G)}, and it is called a fusion class of order i if a has order i. This paper gives a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where:(i) G has at most two fusion classes of order i for every i (23 examples); and(ii) any two elements of G of the same order are fused or inverse-fused.The examples in case (ii) are: A(5), A(6), L(2)(7), L(2)(8), L(3)(4), Sz(8), M(11) and M(23) An application is given concerning isomorphisms of Cayley graphs.