On the number of conjugacy classes of finite nilpotent groups.
E. Landau has shown that there is a finite number of finite groups \(G\) with given class number \(k(G)\). Some estimates for \(|G|\) were obtained; all these estimates are not attained even for groups of modest orders. Problem 20 from \textit{Ya. G. Berkovich} and \textit{E. M. Zhmud'} [Characters of finite groups. Part 2. Translations of Mathematical Monographs 181. Providence: AMS (1998; Zbl 0934.20009)] offers to prove that, for any finite group \(G\), one has \(k(G)\geq\lambda(G)\), where \(\lambda(G)\) is the number of prime factors of \(|G|\) (for example, \(\lambda(\text{PSL}(2,7))=5\)). That estimate is attained for M\(_{22}\) and \(\text{PSL}(3,4)\). In this note the following theorem is proved (the proof is fairly involved). Theorem 1.1. There exists a (explicitly computable) constant \(C>0\) such that every finite nilpotent group \(G\) of order \(n\geq 5\) satisfies \[ k(G)>C\cdot\frac{\log_2\log_2n}{\log_2\log_2\log_2n}\cdot\log_2n.\tag{*} \] The author has noticed that \(C\) can be taken equal to \(10^{-4}\). -- To see that the estimate in (*) is far from the real one, it suffices to take \(|G|=2^{256}\).
scientific article; zbMATH DE number 5907549
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