Regular abelian coverings of cyclic hypermaps

Abstract

The foundations of this work lie in the use of representation theory to find lattices of regular hypermaps H arising as elementary abelian coverings of some regular hypermap Ji such that Aut(Tt) = CR. It is assumed that Aut(7f) = C™ X CR a spUt extension of an elementary abelian group with cyclic complement. Examples of this situation are found in [2] where regular orientable imbeddings of the complete graph Kq, q = pe with p a prime integer, were classified and enumerated. Initial investigations are of maps obtained from regular orientable imbeddings of certain highly symmetrical subgraphs Kq of Kq. Results pertaining to the number and valency of vertices, edges, faces, Petrie polygons, the genus, the automorphism group and reflexibility are found to be similar to those obtained in [2]. Subsequently, a description is given of the general construction. Determination of the lattice of coverings H of H is found to be dependent upon the signature of the map-subgroup T of H. Results are obtained for the cases where Tab is a torsion group and where F"6 is a free abelian group. Hypermap operations induced by outer automorphisms of the triangle group A(R, R, oo) are also considered here. The hypermaps are regular of type (R, R,p) and, once again, have automorphism group C™ XI CR. It was found that when n = 1 these hypermaps lie in a single orbit under the group of hypermap operations, whilst when n = 2 they have orbit length <p{R)2 in case R\p - 1 and length (p(R)2/2 otherwise.

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