For a positive integer m and a finite group G, let

$$\begin{aligned} u_{2'}(G,m)=\frac{\sum _{\chi \in \mathrm{Irr}_{2'}(G)}\chi (1)^{m}}{\sum _{\chi \in \mathrm{Irr}_{2'}(G)}\chi (1)^{m-1}}, \end{aligned}$$

where \(\mathrm{Irr}_{2'}(G)\) denotes the set of all complex irreducible characters of G of odd degrees. The Thompson’s theorem on character degrees states that if \(u_{2'}(G,m)=1\), then G is 2-nilpotent. In this paper, we prove that if

$$\begin{aligned} u_{2'}(G,m)< \frac{3+3^{m}}{3+3^{m-1}}, \end{aligned}$$

then G is 2-nilpotent. This is a strengthened version of Thompson’s theorem in terms of \(u_{2'}(G,m)\).

中文翻译：

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奇数度的性格和汤普森性格度定理

对于正整数m和有限群G，令

$$ \ begin {aligned} u_ {2'}（G，m）= \ frac {\ sum _ {\ chi \ in \ mathrm {Irr} _ {2'}（G）} \ chi（1）^ { m}} {\ sum _ {\ chi \ in \ mathrm {Irr} _ {2'}（G）} \ chi（1）^ {m-1}}，\ end {aligned} $$

其中\（\ mathrm {Irr} _ {2'}（G）\）表示奇数度的G的所有复数不可约性字符的集合。关于字符度的汤普森定理指出，如果\（u_ {2'}（G，m）= 1 \），则G为2幂等。在本文中，我们证明

$$ \ begin {aligned} u_ {2'}（G，m）<\ frac {3 + 3 ^ {m}} {3 + 3 ^ {m-1}}，\ end {aligned} $$

那么G是2幂零的。这是根据\（u_ {2'}（G，m）\）的Thompson定理的增强版本。