On the asymptotic behaviour of \(\rho(n)\)
For a positive integer \(n\), denote by \(\sigma(n)\) the number of different prime divisors of \(n\). For a finite group \(G\), denote by \(\sigma(G)\) the maximum of \(\sigma(\text{ord}(g))\) over all \(g\in G\). The paper is concerned with bounding \(\sigma(\#G)\) in terms of \(\sigma(G)\) for finite solvable groups \(G\). Let \(\rho(n)\) be the maximum of \(\sigma(\#G)\) over all finite solvable groups \(G\) such that \(\sigma(G)=n\). By previous work of \textit{W. Bannuscher} and \textit{G. Tiedt} [preprint (1992)] and \textit{J. Zhang} [Proc. Am. Math. Soc. 123, No. 1, 39-44 (1995; Zbl 0816.20024)], it is known that \(\rho(n) \leq {1\over 2} n(n+3)\) for all \(n\). The author improves this bound by showing that \(\rho(n)=O(n\log n)\) as \(n\to\infty\). This result has by now been superseded by later work of the same author [J. Algebra 178, No. 2, 643-652 (1995; see the following review Zbl 0859.20014)], where a linear bound is obtained. See also his doctoral thesis [Zbl 0859.20018].
scientific article; zbMATH DE number 783878