关于具有平凡中心和有界共轭类的有限群的阶
On the order of a finite group with trivial centre and bounded conjugacy classes
AI总结：
本文证明具有平凡中心的有限 n-共轭类群阶至多为 n^{64(log n)^5}，并由此推出任意有限 n-共轭类群对其超中心的指数也有相同上界。
AI中文摘要：
如果一个群的每个共轭类都是有限的且至多包含 $n$ 个元素，则称该群为 $n$-$\bfc$-群。根据 B.~H.~Neumann 的一个定理，此类群的导子群是有限的，且其阶以 $n$ 为界；然而群本身的阶未必有界，正如超特殊 $p$-群所示。我们证明：具有平凡中心的 $n$-$\bfc$-群是有限的，且其阶至多为 $n^{64(\log n)^{5}}$。通过过渡到超中心（hypercentre）的商群，我们推断出无需对群的结构作任何假设：对于任意有限的 $n$-$\bfc$-群 $G$，有 $|G:Z_\infty(G)|<n^{64(\log n)^{5}}$。此处群的有限性仅是为了正确定义超中心所必需的。
英文摘要：
A group is called an $n$-$\bfc$-group if each of its conjugacy classes is finite and contains at most $n$ elements. By a theorem of B.~H.~Neumann the derived subgroup of such a group is finite of $n$-bounded order, whereas the order of the group itself need not be bounded, as extraspecial $p$-groups show. We prove that an $n$-$\bfc$-group with trivial centre is finite of order at most $n^{64(\log n)^{5}}$. Passing to the quotient by the hypercentre, we deduce that no assumption on the structure of the group is needed at all: for an arbitrary finite $n$-$\bfc$-group $G$ one has $|G:Z_\infty(G)|<n^{64(\log n)^{5}}$. The finiteness of the group is required here only for the correct definition of the hypercentre.
