The multiplicative loops of Jha-Johnson semifields

TL;DR

This paper investigates the structure and properties of the multiplicative loops derived from Jha-Johnson semifields, highlighting their non-automorphic nature and automorphism groups, with bounds on their isotopy classes.

Contribution

It provides new insights into the structure of these loops, including bounds on the number of non-isotopic loops constructed via twisted polynomial rings.

Findings

Multiplicative loops are non-automorphic finite loops.

They have non-trivial automorphism groups and inner mappings.

Upper bounds are established for non-isotopic loops of certain orders.

Abstract

The multiplicative loops of Jha-Johnson semifields are non-automorphic finite loops whose left and right nuclei are the multiplicative groups of a field extension of their centers. They yield examples of finite loops with non-trivial automorphism group and non-trivial inner mappings. Upper bounds are given for the number of non-isotopic multiplicative loops of order $q^{nm}-1$ that are defined using the twisted polynomial ring $K[t;\sigma]$ and a twisted irreducible polynomial of degree $m$, when the automorphism $\sigma$ has order $n$.