Epstein, Michael

A finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to the union of all of the members of C. Such a cover is called minimal if it has the smallest cardinality among all finite covers of G. The covering number of G, denoted by σ(G), is the number of subgroups in a minimal cover of G. Here we determine the covering numbers of the projective special unitary groups U3(q) for q ≤ 5, and give upper and lower bounds for the covering number of U3(q) when q > 5. We also determine the covering number of the McLaughlin sporadic simple group, and verify previously known results on the covering numbers of the Higman-Sims and Held groups.

Any group with a finite noncyclic homomorphic image is a finite union of proper subgroups. Given such
a group G, we define the covering number of G to be the least positive integer m such that G is the
union of m proper subgroups. We present recent results on the determination of the covering numbers
of the alternating groups on nine and eleven letters.