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

TLDR: In this paper, a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where G has at most two fusion classes of order i for every i (23 examples), and any two elements of G of the same order are fused or inversenfused.

Abstract: Elements a,b of a group G are said to be fused if a = bσ and to be inverse-fused if a =(b-1)σ for some σ ϵ Aut(G). The fusion class of a ϵ G is the set {aσ | σ ϵ Aut(G)}, and it is called a fusion class of order i if a has order iThis 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 inversenfused. The examples in case (ii) are: A5, A6,L2(7),L2(8), L3(4), Sz(8), M11 and M23An application is given concerning isomorphisms of Cay ley graphs.