Groups of order p^4 as additive groups of local near-rings - Raievska, I.; Raievska, M. (Ukrains’kyi Matematychnyi Zhurnal, 2024)
UDC 512.6 Near-rings can be considered as a generalization of associative rings. In general, a near-ring is a ring (R,+,·) 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[https://arxiv.org/abs/2303.17567 and 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.