G(k) generated by k-points of copies of the additive group Ga over k contained in G. (By the assumption that G is isotropic over k, the group G(k)+ is nontrivial, and even Zariski dense in G if k is infinite.) Then the quotient group of G(k)+ by its center is simple (as an abstract group). The proof uses Jacques Tits's machinery of BN-pairs. The exceptions for fields of order 2 or 3 are well understood. For k = F2, Tits's simplicity theorem remains valid except when G is split of type A1, B2, or G2, or non-split (that is, unitary) of type A2. For k = F3, the theorem holds except for G of type A1. For a k-simple group G, in order to understand the whole group G(k), one can consider the Whitehead group W(k,G)=G(k)/G(k)+. For G simply connected and quasi-split, the Whitehead group is trivial, and so the whole group G(k) is simple modulo its center. More generally, the Kneser–Tits problem asks for which isotropic k-simple groups the Whitehead group is trivial. In all known examples, W(k,G) is abelian. For an anisotropic k-simple group G, the abstract group G(k) can be far from simple. For example, let D be a division algebra with center a p-adic field k. Suppose that the dimension of D over k is finite and greater than 1. Then G = SL(1,D) is an anisotropic k-simple group. As mentioned above, G(k) is compact in the classical topology. Since it is also totally disconnected, G(k) is a profinite group (but not finite). As a result, G(k) contains infinitely many normal subgroups of finite index. Lattices and arithmetic groups Let G be a linear algebraic group over the rational numbers Q. Then G can be extended to an affine group scheme G over Z, and this determines an abstract group G(Z). An arithmetic group means any subgroup of G(Q) that is commensurable with G(Z). (Arithmeticity of a subgroup of G(Q) is independent of the choice of Z-structure.) For example, SL(n,Z) is an arithmetic subgroup of SL(n,Q). For a Lie group G, a lattice in G means a discrete subgroup Γ of G such that the manifold G/Γ has finite volume (with respect to a G-invariant measure). For example, a discrete subgroup Γ is a lattice if G/Γ is compact. The Margulis arithmeticity theorem says, in particular: for a simple Lie group G of real rank at