Exponent of a finite group admitting a coprime automorphism of prime order

Abstract

Abstract Let 𝐺 be a finite group admitting an automorphism 𝜙 of prime order 𝑝 such that ( | G | , p ) = 1 (\lvert G\rvert,p)=1 . It is shown that if the fixed-point subgroup for 𝜙 has rank 𝑟 and ( x - 1 ⁢ x ϕ ) e = 1 (x^{-1}x^{\phi})^{e}=\nobreak 1 for each x ∈ G x\in G , then the exponent of [ G , ϕ ] [G,\phi] is ( e , p , r ) (e,p,r) -bounded.