Abstract
We prove that a 2-group has exactly five rational irreducible characters if and only if it is dihedral, semidihedral or generalized quaternion. For an arbitrary prime p, we say that an irreducible character χ of a p-group G is “almost rational” if ℚ(χ) is contained in the cyclotomic field ℚ p , and we write ar(G) to denote the number of almost-rational irreducible characters of G. For noncyclic p-groups, the two smallest possible values for ar(G) are p 2 and p 2 + p − 1, and we study p-groups G for which ar(G) is one of these two numbers. If ar(G) = p 2 + p − 1, we say that G is “exceptional”. We show that for exceptional groups, |G: G′| = p 2, and so the assertion about 2-groups with which we began follows from this. We show also that for each prime p, there are exceptional p-groups of arbitrarily large order, and for p ≥ 5, there is a pro-p-group with the property that all of its finite homomorphic images of order at least p 3 are exceptional.
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The second and third authors were supported by the Spanish Ministry of Education, Grants MTM2010-61161 and MTM2008-06680-C02-02, respectively. The third author was also supported by FEDER funds and the Basque Government under Grant IT-460-10.
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Isaacs, I.M., Navarro, G. & Sangroniz, J. p-groups having few almost-rational irreducible characters. Isr. J. Math. 189, 65–96 (2012). https://doi.org/10.1007/s11856-011-0153-y
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DOI: https://doi.org/10.1007/s11856-011-0153-y