The products of conjugacy classes in some infinite simple groups (Q1060280)

Let S be the group of all permutations of a set of cardinality \(\aleph_{\nu}\) and \(S^{\nu}\) its subgroup of permutations moving less than \(\aleph_{\nu}\) elements. Set \(H_{\nu}=S/S^{\nu}\). It is known that \(H_{\nu}\), \(\nu >0\), is simple and that for each nonunit conjugacy class C, \(C^ 2=H_{\nu}\). The only finite simple group with this property is \(J_ 1\), Janko's small group. The Janko group \(J_ 1\), also satisfies \(C_ 1\subseteq C_ 2\cdot C_ 3\) for any three nonunit conjugacy classes \(C_ 1\), \(C_ 2\), \(C_ 3\). At present the groups \(H_{\nu}\), \(\nu >0\), are the only known groups which satisfy \(C^ 2=H_{\nu}\), and for which \(C_ 1\subseteq C_ 2\cdot C_ 3\) fails for certain sets of nonunit conjugacy classes \(C_ 1\), \(C_ 2\), \(C_ 3\). The present paper is concerned mainly with the determination of those sets of conjugacy classes \(C_ 1\), \(C_ 2\), \(C_ 3\) in \(H_{\nu}\) for which \(C_ 1\subseteq C_ 2\cdot C_ 3\) does not hold. The author investigates arbitrary products of classes in \(H_{\nu}\), \(\nu >0\). The notation needed to state the main results is too complicated for a brief review. It should be mentioned, however, that the author uses some of the theory of planar Eulerian graphs to prove the main theorems.