Finite groups with an automorphism of large order

Abstract

Abstract Let G be a finite group, and assume that G has an automorphism of order at least ρ ⁢ | G | {\rho|G|} , with ρ ∈ ( 0 , 1 ) {\rho\in(0,1)} . We prove that if ρ > 1 / 2 {\rho>1/2} , then G is abelian, and if ρ > 1 / 10 {\rho>1/10} , then G is solvable, whereas in general, the assumption implies [ G : Rad ( G ) ] ≤ ρ - 1.78 {[G:\operatorname{Rad}(G)]\leq\rho^{-1.78}} , where Rad ⁡ ( G ) {\operatorname{Rad}(G)} denotes the solvable radical of G. We also prove analogous results for a larger class of self-transformations of finite groups, so-called bijective affine maps. Furthermore, we provide two results of independent interest: an upper bound on element orders in the holomorph of a finite group, and that every bijective affine map of a finite semisimple group has a cycle of length equal to the order of the map, extending a theorem of Horoševskiĭ.