The greatest power of a prime p dividing the natural number n will be denoted by \(n_p\). For a set of primes \(\pi \) and a natural number n we will denote \(n_{\pi }=\prod _{p\in \pi }n_p\). Let G be a finite group with trivial center, and \(p,q>5\) be distinct prime divisors of |G|. We prove that if for every nonunity conjugacy classes size \(\alpha \), it is true that \(\alpha _{\{p,q\}}\in \{p^n,q^m,p^nq^m\}\), where n and m depend only on p and q, then \(|G|_{\{p,q\}}=p^nq^m\), and \(C_G(g)\cap C_G(h)=1\) for every p-element g and every q-element h.

中文翻译：

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限制共轭类大小集的有限群

质数p除以自然数n的最大功效将由\（n_p \）表示。对于一组素数\（\ pi \）和自然数n，我们将表示\（n _ {\ pi} = \ prod _ {p \ in \ pi} n_p \）。令G为具有平凡中心的有限群，\（p，q> 5 \）为|的不同素数。G |。我们证明，如果对于每个非统一共轭类大小\（\ alpha \），在\ {p ^ n，q ^ m，p ^ nq中的\（\ alpha _ {\ {p，q \}}} \ ^ m \} \），其中n和m仅取决于p和q，则每个p元素g和\（| G | _ {\ {p，q \}} = p ^ nq ^ m \）和\（C_G（g）\ cap C_G（h）= 1 \）每q个元素h。