On almost p-rational characters of p'-degree

2021/04/07 by Hung, Nguyen Ngoc, Malle, Gunter, Maróti, Attila · 2 citations
#20D10 #20D20 #20E45 #FOS: Mathematics #Group Theory (math.GR) #Primary 20C15 #Representation Theory (math.RT) #Secondary 20D05

paper · doi:10.48550/arxiv.2104.02994

Abstract

Let p be a prime and G a finite group. A complex character of G is called almost p-rational if its values belong to a cyclotomic field ℚ(e^2πi/n) for some n∈ ℤ⁺ prime to p or precisely divisible by p. We prove that, in contrast to usual p-rational characters, there are always "many" almost p-rational irreducible characters in finite groups. We obtain both explicit and asymptotic bounds for the number of almost p-rational irreducible characters of G in terms of p. In fact, motivated by the McKay-Navarro conjecture, we obtain the same bound for the number of such characters of p'-degree and prove that, in the minimal situation, the number of almost p-rational irreducible p'-characters of G coincides with that of N_G(P) for P\inSyl_p(G). Lastly, we propose a new way to detect the cyclicity of Sylow p-subgroups of a finite group G from its character table, using almost p-rational irreducible p'-characters and the blockwise refinement of the McKay-Navarro conjecture.

Cited by

• Characters and the Generation of Sylow 3-Subgroups For Almost Simple Groups
• Principal blocks, irreducible restriction, fields and degrees

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• On almost p-rational characters in principal blocks
• The continuity of p-rationality and a lower bound for p'-degree characters of finite groups
• The McKay Conjecture on character degrees
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