2-covering numbers of some finite solvable groups

Abstract

A 2-covering for a finite group $G$ is a set of proper subgroups of $G$ such that every pair of elements of $G$ is contained in at least one subgroup in the set. The minimal number of subgroups needed to 2-cover a group $G$ is called the 2-covering number and denoted by $\sigma_2(G).$ In \cite{gk} it is conjectured that if $G$ is solvable and not 2-generated, then $\sigma_2(G)=1+q+q^2,$ where $q$ is a prime power. We disprove this conjecture.