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Nonsolvable groups with no prime dividing three character degrees. (English) Zbl 1246.20006
Authors’ summary: “We consider nonsolvable finite groups \(G\) with the property that no prime divides at least three distinct character degrees of \(G\). We first show that if \(S\leqslant G\leqslant\operatorname{Aut}\,S\), where \(S\) is a nonabelian finite simple group, and no prime divides three degrees of \(G\), then \(S\cong\text{PSL}_2(q)\) with \(q\geqslant 4\). Moreover, in this case, no prime divides three degrees of \(G\) if and only if \(G\cong\text{PSL}_2(q)\), \(G\cong\text{PGL}_2(q)\), or \(q\) is a power of 2 or 3 and \(G\) is a semi-direct product of \(\text{PSL}_2(q)\) with a certain cyclic group. More generally, we give a characterization of nonsolvable groups with no prime dividing three degrees. Using this characterization, we conclude that any such group has at most 6 distinct character degrees, extending to the nonsolvable case the analogous earlier result of D. Benjamin for solvable groups.”
Indeed, that summary covers most of the paper. In order to establish what the authors are aiming at, they work with a lot of details with regard to character degrees and its graphs. The reader should take notice of all the rich details in this paper. Implicit the authors need much specific facts from number theory which they work out very nicely.

MSC:
20C15 Ordinary representations and characters
20D60 Arithmetic and combinatorial problems involving abstract finite groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20D06 Simple groups: alternating groups and groups of Lie type
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References:
[1] Benjamin, D., Coprimeness among irreducible character degrees of finite solvable groups, Proc. amer. math. soc., 125, 2831-2837, (1997) · Zbl 0889.20004
[2] Bianchi, M.; Chillag, D.; Lewis, M.L.; Pacifici, E., Character degree graphs that are complete graphs, Proc. amer. math. soc., 135, 671-676, (2007) · Zbl 1112.20006
[3] Carter, R.W., Finite groups of Lie type: conjugacy classes and complex characters, (1985), Wiley New York · Zbl 0567.20023
[4] Conway, J.H.; Curtis, R.T.; Norton, S.P.; Parker, R.A.; Wilson, R.A., Atlas of finite groups, (1984), Oxford University Press London · Zbl 0568.20001
[5] Dornhoff, L., Group representation theory, part A: ordinary representation theory, (1971), Marcel Dekker New York · Zbl 0227.20002
[6] Huppert, B., Endliche gruppen I, (1983), Springer-Verlag Berlin
[7] Huppert, B., Character theory of finite groups, (1998), Walter de Gruyter Berlin · Zbl 0932.20007
[8] Isaacs, I.M., Character theory of finite groups, (1976), Academic Press San Diego · Zbl 0337.20005
[9] Lewis, M.L., An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky mountain J. math., 38, 175-211, (2008) · Zbl 1166.20006
[10] Lewis, M.L.; Moret√≥, A.; Wolf, T., Non-divisibility among character degrees, J. group theory, 8, 561-588, (2008) · Zbl 1097.20011
[11] Li, T.; Liu, Y.; Song, X., Finite nonsolvable groups whose character graphs have no triangles, J. algebra, 323, 8, 2290-2300, (2010) · Zbl 1207.20005
[12] Manz, O.; Staszewski, R.; Willems, W., On the number of components of a graph related to character degrees, Proc. amer. math. soc., 103, 31-37, (1988) · Zbl 0645.20005
[13] Manz, O.; Wolf, T.R., Representation theory of solvable groups, (1993), Cambridge University Press Cambridge
[14] McVey, J.K., Prime divisibility among degrees of solvable groups, Comm. algebra, 32, 3391-3402, (2004) · Zbl 1077.20009
[15] Schmid, P., Extending the Steinberg representation, J. algebra, 150, 254-256, (1992) · Zbl 0794.20022
[16] Simpson, W.A.; Frame, J.S., The character tables for \(\operatorname{SL}(3, q)\), \(\operatorname{SU}(3, q^2)\), \(\operatorname{PSL}(3, q)\), \(\operatorname{PSU}(3, q^2)\), Canad. J. math., 25, 486-494, (1973) · Zbl 0264.20010
[17] Steinberg, R., The representations of \(\operatorname{GL}(3, q)\), \(\operatorname{GL}(4, q)\), \(\operatorname{PGL}(3, q)\), and \(\operatorname{PGL}(4, q)\), Canad. J. math., 3, 225-235, (1951) · Zbl 0042.25602
[18] Suzuki, M., On a class of doubly transitive groups, Ann. of math., 75, 105-145, (1962) · Zbl 0106.24702
[19] Wu, Y.-T.; Zhang, P., Finite solvable groups whose character graphs are trees, J. algebra, 308, 2, 536-544, (2007) · Zbl 1118.20012
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