Abstract
Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito–Michler Theorem asserts that if a prime p does not divide the degree of any χ \(\in\) Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a real-valued version of this theorem, where instead of Irr(G) we only consider the subset Irrrv(G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irrrv(G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained.
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Part of this paper was done while the second author visited the Mathematics Department of the Università di Firenze. He would like to thank the Department for its hospitality. The authors are also grateful to F. Lübeck for helping them with some computer calculations.
The research of the first author was partially supported by MIUR research program “Teoria dei gruppi ed applicazioni”. This research of the second author was partially supported by the Spanish Ministerio de Educación y Ciencia proyecto MTM2004-06067-C02-01. The third author gratefully acknowledges the support of the NSA and the NSF.
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Dolfi, S., Navarro, G. & Tiep, P.H. Primes dividing the degrees of the real characters. Math. Z. 259, 755–774 (2008). https://doi.org/10.1007/s00209-007-0247-8
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DOI: https://doi.org/10.1007/s00209-007-0247-8