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Coprimeness among irreducible character degrees of finite solvable groups. (English) Zbl 0889.20004
Let \(G\) be a finite nonabelian solvable group. Let \(\text{cd}(G)\) denote the set of the degrees of the irreducible complex characters of \(G\). Let \(k\) be the smallest positive integer with the following property: each set of \(k\) distinct elements of \(\text{cd}(G)\) is setwise relatively prime. It is proved that the cardinality of \(\text{cd}(G)\) is bounded above by a quadratic function of \(k\). The examples given show that the bound obtained in the paper is the best possible for \(k\leq 4\). Further examples show that, in general, the best bound can be no smaller than the linear bound \(3(k-1)\).

20C15 Ordinary representations and characters
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
Full Text: DOI
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