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 Spath, 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.

中文翻译：

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有限群的奇次表示数的下限

令 $G$ 是一个有限群，而 $P$ 是 $G$ 的 Sylow $2$-子群。我们根据 $P$ 的阿贝尔化的大小获得了 $G$ 的奇次不可约复数表示的数量的渐近和显式边界。为此，我们一方面利用 Malle 和 Spath 对质数 2 的 McKay 猜想的最新证明，另一方面，证明了奇数的半直积的类数的下界-order group 作用于 abelian $2$-group。