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Algebra

 

Groups of order p^4 as additive groups of local nearrings


Prelegent: Iryna Raievska, Maryna Raievska

2024-01-18 12:15

It is not true that any finite group is the additive group of a nearring with identity. Therefore it is important to determine such groups and to classify some classes of nearrings with identity on these groups, for example, local nearrings. In [I. Raievska, M. Raievska, 2023] it was shown that on each group of order p^3 with p>2 there exists a local nearring. Moreover, lower bounds for the number of local nearrings on groups of order p^3 are obtained. It is established that on each non-metacyclic non-abelian or metacyclic abelian group of order p^3 there exist at least p+1 non-isomorphic local nearrings. In [I. Raievska, M. Raievska, 2021] it is proved that, up to an isomorphism, there exist at least p local nearrings on elementary abelian additive groups of order p^3, which are not nearfields.
The next natural step is to investigate groups of order p^4 as the additive groups of local nearrings. We consider groups of  nilpotency class 2 of order p^4 which are the additive group of local nearrings [I. Raievska, M. Raievska, 2023]. It is shown that, for odd p, out of 6 such groups 4 are the additive groups of local nearrings. Next, we investigate groups of nilpotency class 3 of order p^4 as additive groups of local nearrings in [I. Raievska, M. Raievska, 2023]. It is shown that for p>3 there exist local nearrings on two of the four non-isomorphic groups of nilpotency class 3 of order p^4.