The near-rings hosted by a class of groups

Abstract

Clay (3), Johnson (5) and Krimmel (6) have each considered the near-rings with identity on dihedral groups. Krimmel actually generalised the class of dihedral groups and investigated the class of finite non-abelian groups with a cyclic normal subgroup of prime index: we shall call this class . Krimmel considered the near-rings with identity that might be denned on members of and he determined the subclass of groups in which support near-rings of this kind. He also managed to calculate the number of non-isomorphic near-rings involved for certain cases. His methods were essentially combinatorial, and his results were expressed in terms of various integers which characterised the individual members of . Certain features of this work led us to investigate the structure of near-rings on members of from a more algebraic point of view and thereby to complete and extend Krimmel's programme. Part of the work in this paper formed the basis of the second author's thesis (7). We should like to thank Dr. J. Krimmel for permission to include some of his results, and Dr. J. Meldrum who detected an error in our original formulation of Theorem 7.1.