Finite groups with small conjugacy classes
Let \(G\) be a finite group. For \(g \in G\) let \(\sigma(g)\) denote the set of different primes dividing the class length \(| G: C_ G(g)|\). Put \(\sigma^*(G) = \max \{| \sigma(g)|\;g\in G\}\) and \(\rho^*(G) = \bigcup_{g \in G} \sigma(g)\). Huppert has conjectured that for solvable groups \(G\) always (I) \(| \rho^*(G)| \leq 2\sigma^* (G)\). Meanwhile this conjecture has attracted several authors and it has been verified for \(\sigma^*(G) \leq 2\). In a very surprising paper [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 2, 111-113 (1991; Zbl 0743.20018)] the author confirmed the inequality (I) for every perfect group. In the paper under review the author focuses on arbitrary groups with \(\sigma^*(G) = 2\) and solvable groups with \(\sigma^*(G) = 3\). It turns out that (I) holds true and equality occurs only for metabelian groups. Quite remarkable is the fact that the non-solvable groups with \(\sigma^*(G) = 2\) coincide exactly with those for which \(\sigma(G) = 1\) which have been classified by \textit{O. Manz} [J. Algebra 96, 114-119 (1985; Zbl 0567.20004)]. Here \(\sigma(G)\) is the maximum number of different primes dividing the degree of an irreducible character of \(G\). In analogy to the above, Huppert also conjectured that for solvable groups \(| \rho(G)| \leq 2\sigma(G)\) where \(\rho(G) = \{p \mid p\) prime, \(p \mid \chi(1)\) for some \(\chi \in \text{Irr}(G)\}\).
scientific article; zbMATH DE number 599657
scientific article