Nonsolvable groups with few character degrees
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Nonsolvable Groups with No Prime Dividing Three Character Degrees
Throughout this note, G will be a finite group, Irr(G) will be the set of irreducible characters of G, and cd(G) will be the set of character degrees of G. We consider groups where no prime divides at least three degrees in cd(G). Benjamin studied this question for solvable groups in [1]. She proved that solvable groups with this property satisfy |cd(G)| 6 6. She also presented examples to show...
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For a finite group G, the character degree graph ∆(G) is the graph whose vertices are the primes dividing the degrees of the ordinary irreducible characters of G, with distinct primes p and q joined by an edge if pq divides some character degree of G. We determine all graphs with four vertices that occur as ∆(G) for some nonsolvable group G. Along with previously known results on character degr...
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A finite group G is odd-square-free if no irreducible complex character of G has degree divisible by the square of an odd prime. We determine all odd-square-free groups G satisfying S 6 G 6 Aut(S) for a finite simple group S. More generally, we show that ifG is any nonsolvable odd-square-free group, then G has at most two nonabelian chief factors and these must be simple odd-square-free groups....
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Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph ∆(G) is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ∈ cd(G). It is shown using the degree graphs of the finite simple groups that if G is a nonsolvable group, then the diameter of ∆(G) is at...
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Throughout this paper, G is a finite group and Irr(G) is the set of irreducible characters of G. We are particularly interested in the values these characters take on the identity of G. If χ ∈ Irr(G), then χ(1) is the degree of χ. The set of all degrees for G is written cd(G) = {χ(1) |χ ∈ Irr(G)}. In recent years, there has been much interest in finding connections between the structure of a fi...
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