Abstract
Let G be a finite group and P a Sylow 2-subgroup of G. We obtain both asymptotic and explicit bounds for the number of odd-degree irreducible complex representations of G in terms of the size of the abelianization of P. To do so, we, on one hand, make use of the recent proof of the McKay conjecture for the prime 2 by Malle and Späth, and, on the other hand, prove lower bounds for the class number of the semidirect product of an odd-order group acting on an abelian 2-group.
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We are grateful to the referee for suggesting us to look at an asymptotic bound for the number of odd-degree representations of a finite group. Theorem 1.1 and its proof arise from his/her suggestions and ideas. This work was initiated when the first author visited the Department of Mathematics at Texas State University during Fall 2017, and he would like to thank the department for its hospitality. The third author is partially supported by a grant from the Simons Foundation (No 499532).
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Hung, N.N., Keller, T.M. & Yang, Y. A lower bound for the number of odd-degree representations of a finite group. Math. Z. 298, 1559–1572 (2021). https://doi.org/10.1007/s00209-020-02660-z
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DOI: https://doi.org/10.1007/s00209-020-02660-z