# Local nearrings on finite non-abelian $2$-generated $p$-groups

## TL;DR

This paper classifies finite non-abelian, non-metacyclic 2-generated p-groups of nilpotency class 2 with cyclic commutator subgroup that serve as additive groups of local nearrings, revealing the structure of their non-invertible elements.

## Contribution

It provides a detailed description of such p-groups as additive groups of local nearrings, including the index of non-invertible elements subgroup.

## Key findings

- Subgroup of non-invertible elements has index p in the additive group.
- Characterization of these p-groups as additive groups of local nearrings.
- Structural properties of the groups with cyclic commutator subgroup.

## Abstract

Finite non-abelian non-metacyclic $2$-generated $p$-groups (${p>2}$) of nilpotency class $2$ with cyclic commutator subgroup which are the additive groups of local nearrings are described. It is shown that the subgroup of all non-invertible elements is of index $p$ in its additive group.