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⁢xϕ)e=1(x^{-1}x^{\phi})^{e}=\nobreak 1 for each x∈Gx\in G, then the exponent of [G,ϕ][G,\phi] is (e,p,r)(e,p,r)-bounded.

中文翻译：

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允许素数阶互质自同构的有限群的指数

令𝐺为一个允许素数阶为auto的自同构𝜙的有限组，使得（| G |，p）= 1（\ lvert G \ rvert，p）= 1。结果表明，如果𝜙的不动点子组的秩为𝑟，并且（x-1⁢xϕ）e = 1（x ^ {-1} x ^ {\ phi}）^ {e} = \ nobreak 1 x∈Gx\ in G，则[G，ϕ] [G，\ phi]的指数为（e，p，r）（e，p，r）有界。