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\begin{document}$$\mathcal{M}$$\end{document} be an orientably regular (resp. regular) map with the number n vertices. By G+\documentclass[12pt]{minimal}
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\begin{document}$$G^+$$\end{document} (resp. G) we denote the group of all orientation-preserving automorphisms (resp. all automorphisms) of M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{M}$$\end{document}. Let π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} be the set of prime divisors of n. A Hall π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}-subgroup of G+\documentclass[12pt]{minimal}
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\begin{document}$$G^+$$\end{document}(resp. G) is meant a subgroup such that the prime divisors of its order all lie in π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} and the primes of its index all lie outside π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}. It is mainly proved in this paper that (1) suppose that M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{M}$$\end{document} is an orientably regular map where n is odd. Then G+\documentclass[12pt]{minimal}
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\begin{document}$$G^+$$\end{document} is solvable and contains a normal Hall π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}-subgroup; (2) suppose that M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{M}$$\end{document} is a regular map where n is odd. Then G is solvable if it has no composition factors isomorphic to PSL(2,q)\documentclass[12pt]{minimal}
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\begin{document}$$\hbox {PSL}(2,q)$$\end{document} for any odd prime power q≠3\documentclass[12pt]{minimal}
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\begin{document}$$q\ne 3$$\end{document}, and G contains a normal Hall π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}-subgroup if and only if it has a normal Hall subgroup of odd order.
