Affiliation:
1. School of Mathematics and Statistics , Carleton University , 1125 Colonel By Drive , Ottawa ON K1S 5B6 , Canada
Abstract
AbstractLet 𝐺 be a group. A functionG→GG\to Gof the formx↦xαgx\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.
Subject
Algebra and Number Theory
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