Pairwise generating and covering sporadic simple groups. (Q984990)

Let \(G\) be a non-cyclic finite group that can be generated by two elements. A subset \(S\) of \(G\) is said to be a pairwise generating set for \(G\) if every distinct pair of elements in \(S\) generates \(G\). The maximal size of a pairwise generating set for \(G\) is denoted by \(\omega(G)\). The minimal number of proper subgroups of \(G\) whose union is \(G\) is denoted by \(\sigma(G)\). Obviously, \(\omega(G)\leq\sigma(G)\) if \(G\) is non-cyclic. In the recent years the precise values or good approximations of \(\omega(G)\) and \(\sigma(G)\) have been obtained for some relevant families of finite simple groups. The methods so far developed are efficient only when the order of \(G\) is ``large''. It should be interesting to develop techniques that work for ``small'' finite simple groups. In this paper the authors provide lower bounds for \(\omega(G)\) and upper bounds for \(\sigma(G)\) whenever \(G\) is one of the 26 sporadic simple groups. In particular the exact value of \(\sigma(G)\) is determined for \(G\) any of the groups \(M_{11}\), \(M_{22}\), \(M_{23}\), \(M_{24}\), \(Fi_{22}\), \(He\), \(Ru\), \(HS\), \(HN\), \(O'N\) and \(Ly\).