Finite solvable tidy groups whose orders are divisible by two primes (Q6656163)

Let \(G\) be a finite group, \(x \in G\) and \(\mathrm{Cyc}_{G}(x)=\{ g\in G \mid \langle x, g \rangle \text{ is cyclic} \}\). In most cases, the subset \(\mathrm{Cyc}_{G}(x)\) is not a subgroup; if this happens for every element \(x \in G\), then \(G\) is said to be a tidy group.\N\NIn the paper under review, the authors classify the tidy \(\{p, q \}\)-groups. Using this characterization they the following:\N\NTheorem 1.1. Let \(G\) be a solvable, tidy group. Then \(G\) has Fitting height at most \(4\) and \(G/F(G)\) has derived length at most \(4\). If \(|G|\) is odd, then \(G\) has Fitting height at most \(3\) and \(G/F(G)\) is abelian or metabelian.