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Let G be a connected algebraic group defined over a field k. Denote by G ( k ) G(k) the group of k-rational points of G. Suppose that A and B are closed subgroups of G defined over k. Then [ A , B ] ( k ) [A,B](k) is not equal to [ A ( k ) , B ( k ) ] [A(k),B(k)] in general. Here [A,B] denotes the group generated by commutators a b a − 1 b − 1 , a ∈ A , b ∈ B ab{a^{ - 1}}{b^{ - 1}},a \in A,b \in B . We say that a field of k of characteristic p is p-closed if given any additive polynomial f ( x ) f(x) in k [ x ] k[x] and any element c in k, there exists an element α \alpha in k such that f ( α ) = c f(\alpha ) = c . Theorem 1. Let G be a connected solvable algebraic group defined over the p-closed field k. Let A and B be closed connected subgroups of G, which are also defined over k, and suppose A normalizes B. T h e n [ A , B ] ( k ) = [ A ( K ) , B ( K ) ] Then\;[A,B]\;(k) = [A(K),B(K)] . 2. If G, A and B are as above and k is only assumed to be perfect then there exists a finite extension k 0 {k_0} of k such that if K is the maximal p-extension of k 0 {k_0} , then [ A , B ] ( K ) = [ A ( K ) , B ( K ) ] [A,B](K) = [A(K),B(K)] .

We conclude an investigation of the problem of strong approximation in algebraic groups. A basic role is played by the proof of the Kneser-Tits conjecture concerning simple, simply connected groups over locally compact fields.

An upper bound is given of the dimension of certain spaces of cusp harmonic forms of arithmetic subgroups Γ of semisimple algebraic groups G in terms of the multiplicities of corresponding irreducible unitary representations of the group GR of real rational points of G in the space 0L2(GR/Γ) of cusp forms.

Algebraic K-theory can be constructed by means of the homotopy groups of the abstract simplicial structure on the group of invertible matrices GL(A) of the ring A. This structure may be naturally taken as two-sidedly invariant. Of basic interest is the multiplication in the functor so obtained, which for different rings A assumes different aspects.

Many of the more striking elementary properties of abelian varieties generalize to other kinds of algebraic groups, e.g. tori, i.e. direct products of multiplicative groups Gm, and in fact to extensions of tori by abelian varieties. This seems to have been more or less known for some time, but explicit statements are hard to find, a lack we attempt to remedy here. We may mention specifically a recent paper of S. Arima [I1], reading between the lines of which might lead one in this direction. In what follows, the better-known structural facts on algebraic groups are used freely; most of these are in [2] and [4], while the relevant facts on abelian varieties can be found in either [7] or [3].

Let K be any field, and L a separable extension of K of finite degree. L has a structure of vector space over K, and we shall denote this space by V. The space of endomorphisms of V will be denoted by Let x be any element of L, and N(x) the norm of x relative to the extension L/K. N is then a function defined on V with values in K. We shall call N the norm form on V. The multiplicative groups of non-zero elements of K and L will be denoted by K* and L* respectively. Let H be any subgroup of if K*. Then the elements z of L* such that N(z)∈H form a subgroup of L*, which we shall denote by GH. On the other hand the elements s of such that N(sx) = Λ(s)N(x) with Λ(s)∈H for all X∈V, form obviously a subgroup of GL(V), which we shall denote by becomes an algebraic group if H=K* or {1}. In case will mean the group of linear transformations of V leaving semi-invariant the norm form of L/K and in case will mean the group of linear transformations of V leaving invariant the norm form of L/K.

The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received much substance and impetus from the work of Severi on commutative algebraic groups over the complex number field, that of Kolchin, Chevalley, and Borel on algebraic groups of matrices, and especially Weil's research on abelian varieties and algebraic transformation spaces. The main purpose of the present paper is to give a more or less systematic account of a large part of what is now known about general algebraic groups, which may be abelian varieties, algebraic groups of matrices, or actually of neither of these types.