Sharply 2-transitive linear groups Academic Article uri icon

abstract

  • Abstract A group Γ is sharply 2-transitive if it admits a faithful permutation representation that is transitiveand free on pairs of distinct points. Conjecturally, for all such groups there exists a near-field N (i.e. askew field that is distributive only from the left, see Definition 2.10) such that Γ ∼= N × ⋉N. This iswell known in the finite case. We prove this conjecture when Γ < GL n (F) is a linear group, where F isany field with char(F) 6= 2 and that p -char(Γ) 6= 2 (see Definition 2.2). 1 Introduction A sharply k-transitive group is, by definition, a permutation group which acts transitively and freelyon ordered k-tuples of distinct points. Quite early on it was realized that k is very limited; in his 1872paper, [10], Jordan proved that finite sharply k-transitive groups with k ≥ 4 are either symmetric,alternating or one of the Mathieu group M 11 ,M 12 . In the infinite case it was proved by J. Tits, in [16],and M. Hall, in [8], that k ≤ 3 for every infinite sharply k-transitive group.Sharply 2-transitive groups attracted the attention of algebraists for many years because they lieon the borderline of permutation group theory and abstract algebraic structures. It is easy to see thatif K is a skew field then the semi-direct product K

publication date

  • January 1, 2014