Fibers of automorphic word maps and an application to composition factors

Abstract

Abstract In this paper, we study the fibers of “automorphic word maps”, a certain generalization of word maps, on finite groups and on nonabelian finite simple groups in particular. As an application, we derive a structural restriction on finite groups G where, for some fixed nonempty reduced word w in d variables and some fixed ρ ∈ ( 0 , 1 ] {\rho\in(0,1]} , the word map w G {w_{G}} on G has a fiber of size at least ρ ⁢ | G | d {\rho|G|^{d}} : No sufficiently large alternating group and no (classical) simple group of Lie type of sufficiently high rank can occur as a composition factor of such a group G.