On the automorphism groups of regular maps

Let 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 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 M is an orientably regular map where n is odd. Then G(+) is solvable and contains a normal Hall pi-subgroup; (2) suppose that M is a regular map where n is odd. Then G is solvable if it has no composition factors isomorphic to PSL(2,q) for any odd prime power q not equal 3, and G contains a normal Hall pi-subgroup if and only if it has a normal Hall subgroup of odd order.