Two Classes Of Regular Maps

In this thesis, we mainly stressed on two problems in regular maps:One is to decide whether a group has regular balanced Cayley maps or not; the other is to determinate which groups can behave as the automorphism groups of Mobius regular maps.For the first problem, we proved that two classes of finite p-groups that have cyclic maximal subgroups don’t have regular balanced Cayley maps. Together with the known results of M. Con-der and Y. Wang, the first problem related to such groups has been solved completely. While to get our results, we used the method of ’quotient groups’ which proved to be more understandable and more simple than the methods they used before.For the second problem, we showed that S6 can’t be the automorphism gorup of Mobius regular maps; in the sense of isomorphism, S7 can be the automorphism group of two Mobius regular maps; and Sg can behave as the automorphism group of three non-isomorphic Mobius regular maps. To get our results, we need to analysis the subgroup structure of Sn. In a sense, this is workable for small n. But, it becomes very hard generally. Anyway, we dealt with Sn for small n under the help of MAGMA.