Groups of nilpotency class $3$ of order $p^4$ as additive groups of local nearrings

Authors

  • I. Raievska University of Warsaw, 2 Stefana Banacha str., 02-097, Warsaw, Poland; Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska str., 01601, Kyiv, Ukraine
  • M. Raievska University of Warsaw, 2 Stefana Banacha str., 02-097, Warsaw, Poland; Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska str., 01601, Kyiv, Ukraine
https://doi.org/10.15330/cmp.17.1.292-301

Keywords:

local nearring, $p$-group, nilpotency class $3$
Published online: 2025-06-30

Abstract

A nearring is a generalization of the notion of a ring in the sense that the addition needs not to be commutative and only one distributive law is assumed. A nearing with identity is called local if the set of all non-invertible elements forms a subgroup of its additive group. However, it is not true that any finite group is an additive group of a nearring with identity. Therefore the determination of the non-abelian finite $p$-groups, which are an additive groups of local nearrings, is an open problem.

We consider groups of nilpotency class $3$ of order $p^4$, which are the additive groups of local nearrings. It is shown that for $p>3$ there exists a local nearring on one of four such groups. Using the well-known system of computer algebra GAP, we construct examples of local nearrings with the specified properties.

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How to Cite
(1)
Raievska, I.; Raievska, M. Groups of Nilpotency Class $3$ of Order $p^4$ As Additive Groups of Local Nearrings. Carpathian Math. Publ. 2025, 17, 292-301.