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The maximal special linear groups which act on translation planes. (English) Zbl 0602.51003

The author studies translation planes \(\pi\) of order \(q^ t\) admitting a collineation group G isomorphic to \(SL(2,p^ r)\) in the translation complement. By a result of D. A. Foulser and the author [J. Algebra 86, 385-406 (1984; Zbl 0527.51012), J. Geom. 18, 122-139 (1982; Zbl 0527.51013)] \(\pi\) is Desarguesian, Hall, Hering, Ott-Schaeffer, the Dempwolff plane of order 16, or one of the two Walker planes of order 25, if \(r=t\) or \(2r=t\). By counting the fixed components of elements whose order is a p-primitive prime divisor of \(p^{2r}-1,\) the author shows that there are no planes for \(t/2<r<t.\)
Reviewer: G.Stroth

MSC:

51A40 Translation planes and spreads in linear incidence geometry
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