On the automorphism groups of regular maps

Abstract

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.