Given a group G, we write x^G for the conjugacy class of G containing the element x. A famous theorem of B. H. Neumann states that if G is a group in which all conjugacy classes are finite with bounded size, then the derived group G′ is finite. We establish the following result.

Let n be a positive integer and K a subgroup of a group G such that ∣x^G∣ ≤ n for each x ∈ K. Let H = 〈K^G〉 be the normal closure of K. Then the order of the derived group H′ is finite and n-bounded.

Some corollaries of this result are also discussed.

中文翻译：

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诺伊曼BFC定理的更强形式

给定的一组G ^，我们写X ^ģ为共轭类的ģ含有元素X。BH诺伊曼（BH Neumann）的一个著名定理指出，如果G是其中所有共轭类别都是有限且有界的组，则导出的组G '是有限的。我们建立以下结果。

让Ñ是正整数和ķ一组的子组G ^使得| X ^ģ |≤ Ñ每个X ∈ ķ。令H = 〈K ^G〉为K的常闭。然后，导出的组H ′的阶为有限且为n界。

还讨论了此结果的一些推论。