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The aim of this paper is to understand subgroups of GL,(Fp), where Fp is the prime field with p elements. To explain our results we need a few simple definitions.Let G be a subgroup of GL, (Fp). Let X = {x~GlxV= l }. Denote by (~ the (connected) algebraic subgroup of GL,, defined over Fv, generated by the oneparameter subgroups t~-*xt= exp (t log x) for all xeX. The (normal) subgroup of G generated by X is denoted by G +.Our main result (see Theorem B, w 3) says that if the prime p is greater than some constant that depends only on n, then G § =G(Fv)+. If G is semi-simple and simply connected, then d(F~)=d(F) + and therefore G+=(~(Fp) in this case. In other words, G § is realized as the group of rational points of the connected algebraic group G.The algebraic groups G thus obtained are not arbitrary -by their very construction they are generated by a finite number of unipotent one-parameter subgroups of (GL,)Fp of the type:where y~M,(Fv) satisfies yP=0. Algebraic subgroups with the above property are said to be exponentially generated. Theorem B also shows that G~---,(~ sets up a one-to-one correspondence between subgroups G of GL,(Fp) such that G =G + and exponentially generated algebraic subgroups of (GL,)F.With G, X and G as above, the Lie algebra of (~ can be obtained directly from G. It is simply the linear span of {logx[x~X}, if p is sufficiently large with respect to n. We denote this linear span by (log G). The Lie subalgebras (log G) of M,(Fp) thus obtained are nilpotently generated, in the sense that they are spanned by their nilpotent elements. We show (TheoremA, w that nil-

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