The multiplicative loops of Jha-Johnson semifields

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 $n$.