The multiplicative loops of Jha-Johnson
                    semifields

Pumpluen, Susanne

doi:10.1090/conm/721/14508

Cited by 1 publication

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“…Another incentive arose from the search for classes of finite loops with non-trivial automorphism groups. Examples can be now obtained as the multiplicative loops of Jha-Johnson semifields [29].…”

Section: Introductionmentioning

confidence: 99%

Automorphisms and isomorphisms of Jha-Johnson semifields obtained from skew polynomial rings

Brown

Pumpluen

Steele

2018

Communications in Algebra

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We study the automorphisms of Jha-Johnson semifields obtained from a right invariant irreducible twisted polynomial f ∈ K[t; σ], where K = F q n is a finite field and σ an automorphism of K of order n, with a particular emphasis on inner automorphisms and the automorphisms of Sandler and Hughes-Kleinfeld semifields. We include the automorphisms of some Knuth semifields (which do not arise from skew polynomial rings).Isomorphism between Jha-Johnson semifields are considered as well.Every finite nonassociative Petit algebra is a Jha-Johnson semifield. These algebras were studied by Wene [35] and more recently, Lavrauw and Sheekey [23].While each Jha-Johnson semifield is isotopic to some such algebra S f it is not necessarily itself isomorphic to an algebra S f . We will focus on those Jha-Johnson semifields which are, and apply the results from [12] to investigate their automorphisms.The structure of the paper is as follows: In Section 1, we introduce the basic terminology and define the algebras S f . Given a finite field K = F q n , an automorphism σ of K of order n with F = Fix(σ) = F q and an irreducible polynomialwe know the automorphisms of the Jha-Johnson semifields S f if n ≥ m − 1 and a subgroup of them if n < m − 1 [12, Theorems 4, 5]. The automorphism groups of Sandler semifields [30] (obtained by choosing n ≥ m and f (t) = t m − a ∈ K[t; σ], a ∈ K \ F ) are particularly relevant: for all Jha-Johnson semifields. We summarize results on the automorphism groups, and give examples when it is trivial and when Aut F (S f ) ∼ = Z/nZ (Theorem 4). Inner automorphisms of Jha-Johnson semifields are considered in Section 2. In Section 3 we consider the special case that n = m and f (t) = t m − a. In this case, the algebras S f are examples of Sandler semifields and also called nonassociative cyclic algebras (K/F, σ, a). The automorphisms of A = (K/F, σ, a) extending id are inner and form a cyclic group isomorphic to ker(N K/F ). We show when Aut F (A) ∼ = ker(N K/F ) and hence consists only of inner automorphisms, when Aut F (A) contains or equals the dicyclic group Dic r of order 4r = 2q + 2, or when Aut F (A) ∼ = Z/(s/m)Z ⋊ q Z/(m 2 )Z contains or equals a semidirect product, where s = (q m − 1)/(q − 1), m > 2 (Theorems 19 and 20). We compute the automorphisms for the Hughes-Kleinfeld and most of the Knuth semifields in Section 4. Not all Knuth semifields are algebras S f , however, the automorphisms behave similarly in all but one case. We compute the automorphism groups in some examples, improving results obtained by Wene [34]. In Section 5 we briefly investigate the isomorphisms between two semifields S f and S g . In particular, we classify nonassociative cyclic algebras of prime degree up to isomorphism.Sections of this work are part of the first and last author's PhD theses [11,33] written under the supervision of the second author.

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Section: Introductionmentioning

confidence: 99%