Regular Cayley Maps Over Minimal Non-abelian Metacyclic Groups

In this thesis, we mainly emphasize on the group actions on maps. We classified the regular balanced Cayley maps of minimal non-abelian metacyclic groups. And, as another group action on maps, we discussed the skew-morphism of Mp(2,1).There are three classes of non-isomorphic minimal non-abelian metacyclic groups:the first one is the quaternion group Q8, the other two classes are denoted by Mp,q(m,r) and Mp(n,m), respectively in this thesis, and one may find the defining relations of them later. The regular balanced Cayley maps of Q8 had been determined before, so we don’t need to say any word about it in this thesis.The group Mp,q(m,r) is also a minimal non-cyclic group. We proved that it has regular balanced Cayley maps if and only if q=2. Moreover, if p-1=2es, (s,2)=1, then it has s non-isomorphic regular balanced cayley maps, where m≥2.While for Mp(n, m) which is a class of p-group, we proved that only when p=2 and m=n or n=m+1, that is only M2(n,n) and M2(n+1,n) have regular balanced Cayley maps. And up to isomorphism, both of these two groups have one regular balanced Cayley map of degree 4.The skew-morphism of a finite group is very important in the sense of determining the existence of regular Cayley maps of this group. We are interested in this problem on finite p-groups. In this thesis, we showed that Mp(2,1) doesn’t have the skew-morphism of order p2.