The products of conjugacy classes in some infinite simple groups

Let H _ν =S/S ^ν, where S is the group of all permutations of a set of cardinality א _ν and S ^v is its subgroup of permutations moving less than א _ν elements. The infinite simple groups H _ν, ν>0, have covering number two; that is, C ²=H ^ν holds for each nonunit conjugacy class C[M]. Janko's small group J ₁, the only finite simple group with covering number two, satisfies also: {Mathematical expression}. In fact, H _ν (ν>0) are the only groups of covering number two where (*) is known to fail. In this paper we determine arbitrary products of classes in H _ν (ν>0).