Groups of nilpotency class \(2\) of order \(p^4\) as additive groups of local nearrings

Abstract

The paper is devoted to the study of local nearrings (those with identity, for which all non-invertible elements form subgroups of their additive group). A study of such nearrings was first initiated by C. J. Maxson in 1968, and the problem on the determination of the finite \(p\)-groups, which are the additive groups of local nearrings have become one of the most important. Particular cases of this (still unsolved) problem have been studied in many works. In previous papers the authors have shown that, up to isomorphism, there exist at least \(p\) local nearrings on elementary abelian additive groups of order \(p^3\), which are not nearfields, and at least \(p+1\) on each non-metacyclic non-abelian or metacyclic abelian groups of order \(p^3\). In this paper we study the groups of nilpotency class \(2\) of order \(p^4\), which are the additive groups of local nearrings. It is proved that, for odd \(p\), \(4\) out of total number \(6\) of such groups are the additive groups of local nearrings. Explicit examples of the corresponding local nearrings are provided.