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6-BFC groups. (English) Zbl 1167.20325

From the introduction: A group is said to be BFC if its conjugacy classes (of elements) are boundedly finite, and \(n\)-BFC if the largest conjugacy classes have order \(n\). B. H. Neumann proved that a group is BFC if and only if its derived group is finite; the second author showed that the derived group of an \(n\)-BFC group is of order bounded in terms of \(n\). He formulated there the following conjecture, in which \(\lambda_n\) stands for the number of prime factors of \(n\), multiplicities included.
Conjecture. For every \(n\)-BFC group \(G\), the order of \(G'\) is at most \(n^{(1+\lambda(n)}/2)\).
There are nilpotent groups of class 2 and arbitrarily large \(n\) where this bound is achieved. Further, it was proved that the conjecture is true when \(n\) is prime and when \(n=4\), in which case \(G'\) is of order \(4\) or \(8\). There is a wide literature attacking this problem; the best bound achieved so far is that of Segal and Shalev. Vaughan-Lee established the conjecture for nilpotent groups. The smallest value of \(n\) for which the conjecture is not known to be true is \(6\), and the aim of this note is to rectify this by proving the following result.
Theorem. Let G be a 6-BFC group. Then \(G'\) is either \(C_6\) or \(Q_8\).

MSC:

20F24 FC-groups and their generalizations
20E45 Conjugacy classes for groups
20F14 Derived series, central series, and generalizations for groups
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References:

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