Uniquely regular dessins with nilpotent automorphism groups of odd prime power order

Abstract

If a graph is cut along each edge and can be divided into many faces, the graph has been defined to be a map when each face is homomorphic to an open disk and a bipartite map called dessin. A dessin D is regular Aut (D) if action transfer on the edge set. In particular, given a finite group G, a regular dessin is uniquely regular dessin if there is only one isomorphism class of Aut (D) are isomorphic to G. In this paper, for a nilpotent automorphism group of odd prime power order and nilpotency class four, we employ group-theoretical methods to classify these uniquely regular dessins.