@article{Raievska_Raievska_2024, title={Groups of order $p^4$ as additive groups of local near-rings}, volume={76}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/8053}, DOI={10.3842/umzh.v76i5.8053}, abstractNote={<p>UDC 512.6</p> <p>Near-rings can be considered as a generalization of associative rings.<span class="Apple-converted-space"> </span>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.<span class="Apple-converted-space"> </span>A near-ring with identity is called local if the set of all invertible elements forms a subgroup of the additive group.<span class="Apple-converted-space"> </span>In particular, every group is an additive group of some near-ring but not of a near-ring with identity.<span class="Apple-converted-space"> </span>Finding non-Abelian finite $p$-groups that are additive groups of local near-rings is an open problem.</p> <p>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]}.<span class="Apple-converted-space"> </span>It was shown that, for $p>3,$ there exist local near-rings on one of<span class="Apple-converted-space"> </span>four nonisomorphic groups of nilpotency class $3$ of order $p^4$.<span class="Apple-converted-space"> </span>In the present paper, we continue our investigation of the groups of nilpotency class $3$ of order $p^4$.<span class="Apple-converted-space"> </span>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.<span class="Apple-converted-space"> </span><span class="Apple-converted-space"> </span>Examples of local near-rings on this group have been constructed in the GAP computer algebra system.</p>}, number={6}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={RaievskaI. and RaievskaM.}, year={2024}, month={Jul.}, pages={890–906} }