List of transitive finite linear groups
In mathematics, especially in areas of abstract algebra and finite geometry, the list of transitive finite linear groups is an important classification of certain highly symmetric actions of finite groups on vector spaces. The solvable finite 2-transitive groups were classified by Bertram Huppert. The classification of finite simple groups made possible the complete classification of finite doubly transitive permutation groups. This is a result by Christoph Hering. A finite 2-transitive group has a socle that is either a vector space over a finite field or a non-abelian primitive simple group; groups of the latter kind are almost simple groups and described elsewhere. This article provides a complete list of the finite 2-transitive groups whose socle is elementary abelian. Let be a prime, and a subgroup of the general linear group acting transitively on the nonzero vectors of the d-dimensional vector space over the finite field with p elements. Infinite classes There are four infinite classes of finite transitive linear groups. Notice that the exceptional group of Lie type G2(q) is usually constructed as the automorphism groups of the split octonions. Hence, it has a natural representation as a subgroup of the 7-dimensional orthogonal group O(7, q). If q is even,