A linear bound for \(\rho(n)\)
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 \(\sigma(G)=n\). In earlier work of the author [J. Algebra 174, No. 2, 587-598 (1995; see the preceding review Zbl 0859.20013)], it was shown that \(\rho(n)=O(n\log n)\) as \(n\to\infty\), improving the previously known quadratic bound. In the present paper, the author now obtains a linear bound on \(\rho(n)\), namely that \[ \limsup_{n\to\infty}{\rho(n)\over n}\leq 4. \] This is essentially best possible, since in an earlier paper [J. Algebra 170, No. 2, 625-648 (1994; Zbl 0816.20022)], he had shown that \[ \limsup_{n\to\infty}{\rho(n)\over n}\geq 3. \] As to the actual behaviour of \(\rho\), he conjectures that in fact \(\rho(n)=3n\) for all \(n\geq 4\). See also the author's doctoral thesis [Zbl 0859.20018].
scientific article; zbMATH DE number 840722