Isomorphism theory of congruence groups

Abstract

This note is an announcement of four theorems I proved in [5][9] on the isomorphisms of the linear, symplectic, and unitary congruence groups. A sketch of the proofs is given. NOTATION. Let V be an w-dimensional vector space over the field F, n*z2. For crGGLw(F), let cr denote the contragredient of cr (inverse of the transpose). Let ~ denote the natural map of GLn( V) onto PGLn( V) ; for any subset 5 of GLn(V), S is the image of S in PGLn(V) under the map. A transvection r is a linear transformation of determinant one which fixes all vectors of some hyperplane, called the proper hyperplane of r. If r 5 1 then (r — 1) V is a line called the proper line of r. An element f of PGLn(V) is called a (projective) transvection if r is a transvection. The proper line and proper hyperplane of f are defined as the proper line and hyperplane of r.