Solvable groups with a small number of prime divisors in the element orders
Let \(G\) be a solvable finite group and let \(n\) be a natural number. Define \(\sigma(n)\) to be the number of distinct prime divisors of \(n\), \(\sigma(G)\) to be \(\max\{\sigma (o(g)) \mid g\in G\}\) and \(\rho(n) := \max \{\sigma(| G|) \mid G\) is a finite solvable group with \(\sigma(G) = n\}\). The paper under review is centered around the problem of calculating the values of \(\rho(n)\). As a consequence of \textit{G. Higman}'s work on finite groups having all elements of prime power order [J. Lond. Math. Soc. 32, 335-342 (1957; Zbl 0079.03204)], it follows that \(\rho(1) = 2\). Bannuscher-Tiedt and J. Zhang proved independently that \(\rho(2) = 5\) and that in general \(\rho(n) \leq {1\over 2} n(n+3)\). In this paper the author proves that \(\rho(3) = 8\) and establishes a better bound for \(\rho(n)\) if \(n \geq 4 : \rho(n)\leq{1\over2}n(n + 3) - 1\). As a consequence, he is able to show that \(\rho(4)\) is either 12 or 13. The whole argument is based primarily on a careful analysis of a special class of solvable groups, called \(\sigma\)-reduced groups because every solvable group \(G\) contains a \(\sigma\)-reduced subgroup \(U\) with \(\sigma(| U|) = \sigma(| G|)\).
scientific article; zbMATH DE number 703239