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Normal coverings of linear groups. (English) Zbl 1291.20024
For a noncyclic group \(G\), let \(\gamma(G)\) denote the smallest number of conjugacy classes of proper subgroups of \(G\) needed to cover \(G\). Bubboloni, Praeger and Spiga have recently investigated \(\gamma(G)\) in the case that \(G\) is a finite symmetric or alternating group. In this paper the authors give bounds on \(\gamma(G)\), where \(\text{SL}(n,q)\leq G\leq\text{GL}(n,q)\) and \(n>2\). For example they prove that \(n/\pi^2<\gamma(G)\leq (n+1)/2\). They give various alternative bounds and derive explicit formulas for \(\gamma(G)\) in some cases. Let \(\kappa(G)\) be the size of the largest set of conjugacy classes of \(G\) such that any pair of elements from distinct classes generates \(G\). It is clear that whenever \(\gamma(G)\) is defined, we have the inequality \(\kappa(G)\leq\gamma(G)\). There are certain cases in which an upper bound for \(\gamma(G)\) coincides with a lower bound for \(\kappa(G)\). In these cases the authors obtain a precise formula: for example \[ \gamma(G)=\kappa(G)=\Bigl(1-\frac{1}{p}\Bigr)\frac{n}{2}+1\quad\text{ if }n=p^a \] and \[ \gamma(G)=\kappa(G)=\Bigl(1-\frac{1}{p_1}\Bigr)\Bigl(1-\frac{1}{p_2}\Bigr)\frac{n}{2}+2\quad\text{ if }n=p_1^ap_2^b. \]

MSC:
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20G40 Linear algebraic groups over finite fields
20D30 Series and lattices of subgroups
05A15 Exact enumeration problems, generating functions
20E45 Conjugacy classes for groups
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