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On subgroups of $$GL_ n(F_ p)$$. (English) Zbl 0632.20030
Let G be subgroup of $$GL_ n({\mathbb{F}}_ q)$$ where $${\mathbb{F}}_ p$$ is the prime field of p elements, $$X=\{x\in G|$$ $$x^ p=1\}$$. Denote by $$G^+$$ the normal subgroup of G generated by X and denote by $$\tilde G$$ the algebraic subgroup of $$GL_ n$$ generated by the one-parameter subgroups $$t\mapsto x^ t=\exp (t \log x)$$ for all $$x\in X$$. The main result (Theorem B) says that $$G^+=\tilde G({\mathbb{F}}_ p)^+$$ for all primes $$p>c(n)$$. If G is semisimple and simply connected, then $$G^+\cong \tilde G({\mathbb{F}}_ p)$$. An algebraic subgroup $$\tilde G$$ of $$(GL_ n)_{{\mathbb{F}}_ q}$$ is said to be exponentially generated if $$\tilde G$$ is generated by a finite number of one-parameter subgroups exp(ty) where $$y\in M_ n({\mathbb{F}}_ p)$$ satisfies $$y^ p=0$$. Similarly a Lie subalgebra of $$M_ n({\mathbb{F}}_ p)$$ is called nilpotently generated if it is linearly spanned by its nilpotent elements. Theorem A asserts that nilpotently generated Lie subalgebras of $$M_ n({\mathbb{F}})$$ are in one-to-one correspondence with exponentially generated algebraic subgroups of $$(GL_ n)_ F$$ p is sufficiently large. In view of Theorem B, for $$F={\mathbb{F}}_ p$$, they are also in one-to-one correspondence with subgroups G of $$GL_ n({\mathbb{F}}_ p)$$ satisfying $$G=G^+.$$
Let G be a subgroup of GL(V) where V is a vector space of dimension n over $${\mathbb{F}}_ p$$ and let $$<\log G>\subset End V$$ be the linear span of $$\{$$ log $$x|$$ $$x\in X\}$$. The author shows that $$H^ 1(G^+,V)\cong H^ 1(<\log G>,V)$$ if $$p>c_ 2(n)$$ and that $$H^ 1(G,V)=0$$ if $$p>c_ 3(n)$$ and if the action of G on V is semisimple. As an application, the author shows that for a subgroup $$\pi \subset GL_ n({\mathbb{Z}})$$, the two measures of the size of $$\pi$$, i.e. the closure of $$\pi$$ in the profinite group $$GL_ n({\hat {\mathbb{Z}}})$$ and the Zariski-closure of $$\pi$$, are roughly equivalent.
Reviewer: Chen Zhijie

##### MSC:
 20G40 Linear algebraic groups over finite fields 20G10 Cohomology theory for linear algebraic groups 20E07 Subgroup theorems; subgroup growth 20H30 Other matrix groups over finite fields 20F40 Associated Lie structures for groups
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