Simple groups acting on translation planes
The authors study which finite simple groups can act as collineation groups on finite translation planes. Let \(\pi\) be a finite translation plane with kernel GF(q), \(q=p^ k\), \(k\geq 1\), p an odd prime, and order \(q^ d\) (d is the ``dimension of \(\pi\) ''). The authors prove the following. If a finite noncyclic simple group G acts as a collineation group on \(\pi\), then: (i) every involution of G is a Bair involution on \(\pi\) ; (ii) if \(2^ a\) is the order of a maximal elementary abelian 2- subgroup of G, then \(2^ a| d\) and \(2^ a\geq 4\). Now let G be a simple group of collineations of \(\pi\) contained in the linear complement of \(\pi\), which acts irreducibly on \(\pi\). If the Sylow 2-subgroups of G are elementary abelian, then \(p| | G|\) and the representation of G is properly modular. If \(p\nmid | G|\), then G cannot be among 11 simple sporadic groups listed by the authors. Also, if G is a Mathieu group, then it is either \(M_{11}\) or \(M_{23}\).
finite groups acting on translation planes
scientific article; zbMATH DE number 4017743
scientific article