The special linear group is the simply connected group and the projective linear group is the adjoint group of Lie type An. They are distinguished sections of the (reductive) general linear group which certainly is of this type as well (root system). We shall characterize the general linear group as the universal group of type An. Indeed we shall introduce corresponding algebraic groups and finite groups for each Lie type (to indecomposable root systems). Knowledge of the universal group implies knowledge of the related simply connected and adjoint groups; in certain respects the universal group even appears to be better behaved (automorphisms, Schur multiplier, character table).