Finite groups with an affine map of large order

Abstract

AbstractLet 𝐺 be a group. A functionG→GG\to Gof the formx↦xα⁢gx\mapsto x^{\alpha}gfor a fixed automorphism 𝛼 of 𝐺 and a fixedg∈Gg\in Gis called anaffine map of 𝐺. The affine maps of 𝐺 form a group, called the holomorph of 𝐺. In this paper, we study finite groups 𝐺 with an affine map of large order. More precisely, we show that if 𝐺 admits an affine map of order larger than12⁢|G|\frac{1}{2}\lvert G\rvert, then 𝐺 is solvable of derived length at most 3. We also show that, more generally, for eachρ∈(0,1]\rho\in(0,1], if 𝐺 admits an affine map of order at leastρ⁢|G|\rho\lvert G\rvert, then the largest solvable normal subgroup of 𝐺 has derived length at most4⁢⌊log2⁡(ρ−1)⌋+34\lfloor\log_{2}(\rho^{-1})\rfloor+3.