Lower bounds for the number of local nearrings on groups of order \(p^3\) (Q6621965)

The goal of this paper is to get a detailed informtion on finite zerosymmetric near-rings. One knows that their additive group must be a \(p\)-group. If such a near-ring \(N\) has the prime order \(p\), things are easy: \(N\) must be a commutative ring. For the order \(p^2\), a full description of all such near-rings was obtained by a famous result of H. Zassenhaus. For example, one knows that there exist \(p-1\) local near-rings on \(Z_p \times Z_p\) which are not near-fields. For order \(p^3\), things are much harder. The authors have proved before that there are at least \(p\) local near-rings on an elementary abelian group of order \(p^3\). In this paper, they extend this informtion: on each non-metacyclic non-abelian or metacyclic abelian group of order \(p^3\) there are at least \(p+1\) non-isomorphic local near-rings. The main tool is to study the structure of the subgroup of non-invertible elements, going through all the relevant cases (which are a lot).