Let R be a not necessarily commutative local ring, M a free R-module, and π ∈ GL(M) such that B(π) = im(π –1)is a subspace of M. Then π = σ1…σtρ, where σi are simple mappings of given types, ρ is a simple mapping, B(sgr;i) and B(ρ) are subspaces and t ≤ dim B(π).

Using the theory of the Satake diagrams associated with the non-compact simple Lie algebras over the real number field R, we shall construct a family of simple groups over a field K which are called the simple groups associated with the Satake diagrams. The list of these simple groups includes all Chevalley groups and twisted groups, and all simple algebraic groups of adjoint type defined over R if K is the complex number field C (except two types given by Table II′). Furthermore, the simple groups associated with the Satake diagrams of type AIII, BI, DI are identified with the simple groups obtained from the unitary or orthogonal groups of non-zero indices. The quasi-Bruhat decomposition of the “non-split” simple groups associated with the Satake diagrams which are not Chevalley groups or twisted groups will be given in this paper.

Let be a finitely generated subgroup of SL (2, ℱ), where ℱ is the ring; of holomorphic functions on the open unit disc Δ. For each point z0 in Δ we can evaluate all matrix entries of at z0, to obtain a subgroup {z0} of SL (2, ℂ) and a surjective representation → {z0}. If this representation is not faithful, then contains a nontrivial element W such that W evaluated; at z0 is trivial. But W can evaluate to the identity only on a countable subset) of Δ, and there are only countably many choices for W in Consequently there are at most countably many points zk in Δ such that {zk} is not isomorphic to Δ. Our main result can now be stated as follows.

We discuss generalizations of the Lie-Kolchin-Mal'cev theorem. For example we show that if G is a soluble linear group of degree n, then G contains a triangularizable subgroup T whose index in G is bounded by function of n only and such that T is normalized by every automorphism of G normalizing G0, the Zariski connected component of G containing the identity. We also prove that in certain situations at least the index of G0 in G can be bounded in terms of the degree and the ground field.