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Finite linear groups and bounded generation. (English) Zbl 1017.20039

Let \(p\) be a prime and \(F\) a field of characteristic \(p\). The main result of this paper is Theorem A. If \(n\) is a positive integer and \(p\) is sufficiently large, then any finite subgroup of \(\text{GL}(n,F)\) generated by its Sylow \(p\)-subgroups is a product of \(25\) Sylow \(p\)-subgroups. In the case that \(F\) is the prime field, a variation of this result was obtained by E. Hrushovski and A. Pillay [J. Reine Angew. Math. 462, 69-91 (1995; Zbl 0823.12005)]. The proof depends on a recent result of M. Larsen and R. Pink [“Finite subgroups of algebraic groups”, J. Am. Math. Soc. (to appear)] asserting that there are only finitely many simple groups other than Chevalley groups in the natural characteristic having projective representations of dimension at most \(n\). Using the classification of finite simple groups and a recent result of the reviewer [J. Algebra 220, No. 2, 531-541 (1999; Zbl 0941.20001)], \(p\) sufficiently large can be replaced by \(p\geq n+3\) (for \(n\geq 9\)) and this result is best possible. The results of Larsen-Pink (or the classification of finite simple groups) allow one to reduce to the case of Chevalley groups in characteristic \(p\). In that case, the result follows from the Bruhat decomposition.
Using the results mentioned above, the authors also confirm a conjecture of Sury and Platonov that any closed finitely generated subgroup of the direct product of \(\text{SL}(n,\mathbb{Z}_p)\) is boundedly generated (here \(\mathbb{Z}_p\) is the ring of \(p\)-adic integers).
An example is given to show that one cannot replace Sylow \(p\)-subgroup by Abelian \(p\)-subgroup in Theorem A. It is asked whether at least \(G\) (as in Theorem A for \(p\) sufficiently large) is a product of \(g(n)\) Abelian \(p\)-subgroups for some function \(g(n)\). We close this review with an example of Aschbacher and the reviewer showing that this is not true. Note that it is true for \(n=2\).
Let \(q=p^a\). Let \(G\) be the subgroup of \(\text{SL}(3,q)\) consisting of upper triangular unipotent elements such that the \((2,3)\) entry is the \(p\)-th power of the \((1,2)\) entry. This is a group of order \(q^2\). It is straightforward to check that any Abelian subgroup of \(G\) has order at most \(qp\). Thus, \(G\) is a product of \(a\) Abelian subgroups, but no fewer. Since \(a\) can be arbitrary, there is no bound on the number of Abelian subgroups needed to generate a finite subgroup (generated by its elements of order \(p\)) of \(\text{SL}(3,F)\) even if we allow \(p\) to be large. (Also submitted to MR).

MSC:

20G40 Linear algebraic groups over finite fields
20E18 Limits, profinite groups
20F05 Generators, relations, and presentations of groups
20D40 Products of subgroups of abstract finite groups
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