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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 $$3(k-1)$$.

##### MSC:
 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
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##### References:
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