On subgroups ofGL n (F p )

Abstract

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-

Arithmetic groups have rational representation growth

“…The claims in the proposition are trivial if p is bounded. For the proof, we shall assume that p is large enough, and so we can use the results of [18]. Also, the first statement clearly follows from the second, so we prove the second claim.…”

“…Subgroups of GL n (F p ). We review some definitions from [18], and advise the reader to have a copy in hand. We fix a prime number p and a natural number p > n. Let Υ be a subgroup of GL n (F p ).…”

Affine linear sieve, expanders, and sum-product

“…Their proof makes use of the classification of finite simple groups. The treatment of this theorem in [48] does not make use of this classification. To see that part (iii) is true, choose q 1 with q 0 |q 1 and M|q 1 , and so that the following holds: the center Z of G is finite and for p large enough G(F p )/Z(F p ) are distinct finite simple groups.…”

Eigenvalue fields of hyperbolic orbifolds

“…The following result follows from work of Weisfeiler [13] or Nori [10]. For a proof in this setting, see Theorem 2.6 of [6].…”