Let \(T_n\) denote the transformation semigroup consisting of all maps from \([n] \rightarrow [n]\) , where \([n] = \{1,\ldots ,n\}\) . In 2009, Araújo, Mitchell and Schneider gave a classification of subgroups \(G\le S_n\) such that \(\langle G,a\rangle \) is a regular semigroup for all \(a\in T_n\) , showing that for \(n\ge 10\) , the only possibilities for G are \(A_n\) and \(S_n\) . We investigate the analogous problem for linear groups over finite fields, raised by Araújo and Cameron: classify the subgroups of \(\operatorname {GL}_n(q)\) such that \(\langle G,A\rangle \) is a regular semigroup for all matrices A in the matrix semigroup \(M_n(q)\) . We obtain a representation theoretic criterion for such groups G, and exhibit a connection with the action of G on the exterior powers of the underlying space \(\mathbb {F}_q^n\) . As a consequence we prove the existence of several families of subgroups G of \(\operatorname {GL}_n(q)\) for arbitrary n with the property that \(\langle G,A\rangle \) is regular for all A, in contrast with the previously mentioned result for \(T_n\) .
