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Character degree graphs that are complete graphs. (English) Zbl 1112.20006
Let \(G\) be a finite group and let \(\text{cd}(G)\) denote the different integers which occur as the degrees of the irreducible complex characters of \(G\). Let \(\Gamma(G)\) be the graph whose vertex set is \(\text{cd}(G)-\{1\}\). An edge joins two vertices represented by positive integers \(a\) and \(b\) if \(\gcd(a,b)>1\).
The authors prove that if \(\Gamma(G)\) is a complete graph, then \(G\) is solvable.
The proof, as might be anticipated, uses the classification of finite simple groups, and divides itself into three parts. For groups of Lie type, properties of the Steinberg character are used. For alternating groups, two irreducible characters whose degrees are small and relatively prime, and which extend to the symmetric group, are used. For sporadic groups, pairs of irreducible characters of relatively prime degrees, which also extend to the relevant automorphism group, are employed. These are found using the ATLAS.

MSC:
20C15 Ordinary representations and characters
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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