On the nuclei of semifields and Cofman's many subplane problem
type: Item
instance of: scholarly article
title: On the nuclei of semifields and Cofman's many subplane problem
zbMATH Open document ID: 0615.51001
DOI: 10.1007/BF02941605
author: Norman L. Johnson
author: Vikram Jha
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1987-01-01T00:00:00Z
review text: (p,m,n) is a Cofman or C-triple, if an affine translation plane of order \(q^ n\), where \(q=p^ m\), contains more subaffine planes of order q than exist in the Desarguesian plane of the same size, viz., \(q^ n\). \textit{J. W. Freeman} [Geom. Dedicata 9, 267-280 (1980; Zbl 0436.51005)] showed that (p,2M,2) is always a C-triple, thus answering a question of \textit{J. Cofman} [Proc. Internat. Conf. Projective Planes, Washington State Univ. 1973, 59-70 (1973; Zbl 0281.50008)]. In the article being discussed it is conjectured that (p,m,n) is a C-triple iff both m, \(n>1\). The conjecture is proved for even n, and also for many odd values of n: e.g. when \(\gcd (m,n)\) has a square divisor \(>1\). These results are obtained by analysing the conditions under which every semifield, coordinatising a given semifield plane \(\pi\), has GF(q) in the intersection subfield of two of its three seminuclei.
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 12K10
zbMATH DE Number: 3996505
zbMATH Keywords: affine planes
zbMATH Keywords: nuclei of semifields
zbMATH Keywords: translation plane
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02941605
OpenAlex ID: W1964477408
description: scientific article; zbMATH DE number 3996505