A note on the group of projectivities of finite projective planes. (Q1022880)

In his paper [J. Geom. 20, 72--85 (1983; Zbl 0514.51013)], \textit{T. Grundhöfer} has shown that the group of projectivities of a non-Desarguesian projective plane of finite order \(n\) is either the alternating group \(A_{n+1}\), or the symmetric group \(S_{n+1}\), or \(n=23\), and the group of projectivities in this case is the Mathieu group \(M_{24}\). In the present paper the authors show that the latter case cannot occur. A crucial step in their argument is the observation that the Mathieu group \(M_{22}\) does not contain the multiplication group of loop order 22. (For a slightly weaker result, see [\textit{A. Drápal}, J. Algebra 251, No.~1, 256--278 (2002; Zbl 1009.20079)].) To prove this, the authors use also computer calculation.