Character degrees of simple groups (Q1176655)

A set \(S\subseteq\text{Irr}(G)\) is called a covering set if for each prime \(p\) dividing \(| G|\) there exists a \(\chi\in S\) such that \(p|\chi(1)\). If a covering set exists then the covering number \(cn(G)\) is defined by \(cn(G)=\min\{| S| \mid S\hbox{ a covering set}\}\). Note that \textit{G. Michler}'s work [in: Lect. Notes Math. 1178, 129-142 (1986; Zbl 0628.20013)] implies the existence of covering sets for nonabelian simple groups. If the group is of Lie-type, then \(cn(G)\leq 3\), in most cases even \(cn(G)\leq 2\) observed by \textit{O. Manz}, \textit{R. Staszewski} and the reviewer [Proc. Am. Math. Soc. 103, 31-37 (1988; Zbl 0645.20005)]. In the paper under review the authors mainly deal with alternating groups \(A_ n\). For \(n\geq 6\), they prove \(cn(A_ n)\leq 2\). Thus together with inspection of the sporadic groups they obtain Theorem \(A\) which says: For a finite nonabelian simple group \(G\), \(cn(G)\leq 3\). The authors conjecture that \(cn(A_ n)=1\) for \(n\geq 15\). It has been verified up to \(n=100 000\) by computer. Remark: The \(\rho\)-\(\sigma\) relationship \(|\rho(G)|\leq\sigma(G)^ 2+10 \sigma(G)\) mentioned at the beginning of the paper has been improved by \textit{D. Gluck} and \textit{O. Manz} to \(|\rho(G)|\leq 3\sigma(G)+32\) [Bull. Lond. Math. Soc. 19, 431-437 (1987; Zbl 0627.20007)]. Here \(G\) is a solvable group, \(\rho(G)\) the set of primes which divide some irreducible character degree of \(G\) and \(\sigma(G)\) the maximum number of distinct primes which divide any one irreducible character degree.