Lower bounds for the number of local nearrings on groups of order $p^3$

TL;DR

This paper establishes lower bounds on the number of local nearrings that can exist on groups of order p^3, showing at least p+1 such structures for certain groups.

Contribution

It provides the first known lower bounds for local nearrings on groups of order p^3, especially distinguishing between different group structures.

Findings

At least p+1 non-isomorphic local nearrings exist on each non-metacyclic non-abelian group of order p^3.

At least p+1 non-isomorphic local nearrings exist on each metacyclic abelian group of order p^3.

Lower bounds depend on the group's structure, highlighting diversity in local nearring configurations.

Abstract

Lower bounds for the number of local nearrings on groups of order $p^3$ are obtained. On each non-metacyclic non-abelian or metacyclic abelian groups of order $p^3$ there exist at least $p+1$ non-isomorphic local nearrings