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The dimension of a subplane of a translation plane. (English) Zbl 1213.51003

If \(\pi_ 0\) is a subplane of order \(p^ k\) of a desarguesian plane \(\pi\) of order \(p^ t\), then it follows from elementary algebra that \(k\) divides \(t\). The authors show that this result cannot be extended to translation planes. Counterexamples are provided by the translation planes over certain commutative semifields of order \(2^ {5k}\) or \(2^ {7k}, k\) odd, which contain subplanes of order \(4\).

MSC:

51A40 Translation planes and spreads in linear incidence geometry
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Full Text: Euclid