On the covering number of small symmetric groups and some sporadic simple groups

Abstract

Abstract A set of proper subgroups is a cover for a group if its union is the whole group. The minimal number of subgroups needed to cover G is called the covering number of G, denoted by σ ⁢ ( G ) ${\sigma(G)}$ . Determining σ ⁢ ( G ) ${\sigma(G)}$ is an open problem for many nonsolvable groups. For symmetric groups S n ${S_{n}}$ , Maróti determined σ ⁢ ( S n ) ${\sigma(S_{n})}$ for odd n with the exception of n = 9 ${n=9}$ and gave estimates for n even. In this paper we determine σ ⁢ ( S n ) ${\sigma(S_{n})}$ for n = 8 , 9 , 10 , 12 ${n=8,9,10,12}$ . In addition we find the covering number for the Mathieu group M 12 ${M_{12}}$ and improve an estimate given by Holmes for the Janko group J 1 ${J_{1}}$ .