Finite groups with an affine map of large order (Q6110525)

Let \(G\) be a finite group, the holomorph \(\mathrm{Hol}(G)=G \rtimes \mathrm{Aut}(G)\) acts on \(G\) via \(x^{(g,\alpha)}=x^{\alpha}g\). In this paper, the author calls \(x \mapsto x^{\alpha}g\) an affine map of \(G\). The purpose of this article is to study finite groups admitting an affine map of large order. More precisely, he proves that if \(G\) admits an affine map of order larger than \(\frac{1}{2}|G|\), then \(G\) is solvable of derived length at most 3 and, for every \(\rho \in (0,1]\), if \(G\) admits an affine map of order at least \(\rho |G|\), then the largest solvable normal subgroup of \(G\) has derived length at most \(4\lfloor \log_{2}(\rho^{-1}) \rfloor+3\).