On the automorphism groups of regular maps

Let \(\mathcal{M}\) be an orientably regular (resp. regular) map with the number n vertices. By \(G^+\) (resp. G) we denote the group of all orientation-preserving automorphisms (resp. all automorphisms) of \(\mathcal{M}\). Let \(\pi \) be the set of prime divisors of n. A Hall \(\pi \)-subgroup of \(G^+\)(resp. G) is meant a subgroup such that the prime divisors of its order all lie in \(\pi \) and the primes of its index all lie outside \(\pi \). It is mainly proved in this paper that (1) suppose that \(\mathcal{M}\) is an orientably regular map where n is odd. Then \(G^+\) is solvable and contains a normal Hall \(\pi \)-subgroup; (2) suppose that \(\mathcal{M}\) is a regular map where n is odd. Then G is solvable if it has no composition factors isomorphic to \(\hbox {PSL}(2,q)\) for any odd prime power \(q\ne 3\), and G contains a normal Hall \(\pi \)-subgroup if and only if it has a normal Hall subgroup of odd order.