Finite groups in which no two distinct conjugacy classes have the same order (Q1114000)

How many nontrivial groups are described by the title of the paper? Probably only one: \(S_ 3\). - In this note, the author proves \(S_ 3\) is the only finite \(\{\) 2,3\(\}\)-group in which no two distinct conjugacy classes have the same order. The strategy is simple. Show that the existence of certain ``large'' conjugacy classes implies the group is isomorphic to \(S_ 3\) and then show that the class equation insures the presence of these ``large'' classes. This result generalizes two other published papers on the same topic.