Products of conjugacy classes in Chevalley groups. I: Extended covering numbers
Let \(G\) be a group. The `covering number' \(\text{cn}(G)\) is the smallest integer \(m\) such that \(C^m=G\) for every conjugacy class \(C\) of \(G\) which is not contained in any proper normal subgroup of \(G\). The `extended covering number' \(\text{ecn}(G)\) is the smallest integer \(e\) such that the product \(C_1C_2\cdots C_e=G\), whenever \(C_1,C_2,\dots,C_e\) are conjugacy classes of \(G\) not contained in any proper normal subgroup of \(G\). The main result obtained by the authors is the following theorem: There is a constant \(e\) such that for any Chevalley group \(G=G(F)\) (over any field \(F\)), the inequality \(\text{ecn}(G)\leq e\cdot\text{rank}(G)\) holds. This generalizes an earlier result; namely, there is a constant \(c\) such that \(\text{cn}(G)\leq c\cdot\text{rank}(G)\) for every quasi-simple Chevalley group \(G\) [\textit{E. W. Ellers, N. Gordeev, M. Herzog}, Isr. J. Math. 111, 339-372 (1999; Zbl 0941.20050)]. The constant \(e\), as constructed in the proof, is fairly large in general. Better estimates are obtained in special cases, e.g. for \(\text{SL}_n(F)\). The authors also show: Let \(G\) be a finite crystallographic Chevalley group (untwisted or twisted). Then every noncentral conjugacy class of \(G\) has a nonempty intersection with a general Coxeter cell of the Bruhat decomposition of \(G\).
scientific article; zbMATH DE number 1799793
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