Groups of order $p^4$ as additive groups of local near-rings

Abstract

UDC 512.6

Near-rings can be considered as a generalization of associative rings. In general, a near-ring is a ring $(R,+,\cdot)$ in which the operation $``+"$ is not necessarily Abelian and at least one distributive law holds. A near-ring with identity is called local if the set of all invertible elements forms a subgroup of the additive group. In particular, every group is an additive group of some near-ring but not of a near-ring with identity. Finding non-Abelian finite $p$-groups that are additive groups of local near-rings is an open problem.

We considered groups of nilpotency class $2$ and $3$ of order $p^4$ as additive groups of local near-rings in\linebreak {\sf\scriptsize [https://arxiv.org/abs/2303.17567} and {\sf\scriptsize https://arxiv.org/abs/2309.14342]}. It was shown that, for $p>3,$ there exist local near-rings on one of  four nonisomorphic groups of nilpotency class $3$ of order $p^4$. In the present paper, we continue our investigation of the groups of nilpotency class $3$ of order $p^4$. In particular, it is shown that another group of this class is an additive group of a local near-ring and, hence, of a near-ring with identity.  Examples of local near-rings on this group have been constructed in the GAP computer algebra system.