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Linear groups as right multiplication groups of quasifields. (English) Zbl 1301.51004

Translation planes are related to quasifields, besides it is not a one to one correspondence. The right multiplication group of a quasifield is a linear group acting transitively on the nontrivial vectors of a vectorspace. Finite transitive linear groups have been determined by C. Hering [Geom. Dedicata 2, 425–460 (1974; Zbl 0292.20045); J. Algebra 93, 151–164 (1985; Zbl 0583.20003)] and M. Liebeck [Proc. Lond. Math. Soc. 54, 57–72 (1995)]. There are four families and 27 exceptional examples. This paper deals with the exceptional groups and gives a classification of those quasifield whose right multiplication group is one of the exceptional groups.

MSC:

51A40 Translation planes and spreads in linear incidence geometry
05B25 Combinatorial aspects of finite geometries
20N05 Loops, quasigroups

Software:

GAP; Cliquer; GRAPE
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References:

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