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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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