Let α be a coprime automorphism of a group G of prime order and let P be an α-invariant Sylow p-subgroup of G. Assume that $p\notin \pi (C_G(\alpha ))$. Firstly, we prove that G is p-nilpotent if and only if $C_{N_G(P)}(\alpha )$ centralizes P. In the case that G is $Sz(2^r)$ and $PSL(2,2^r)$-free where $r=|\alpha |$, we show that G is p-closed if and only if $C_G(\alpha )$ normalizes P. As a consequence of these two results, we obtain that $G\cong P\times H$ for a group H if and only if $C_G(\alpha )$ centralizes P. We also prove a generalization of the Frobenius p-nilpotency theorem for groups admitting a group of automorphisms of coprime order.

中文翻译：

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自同构的不动点对群结构的影响

设α为素数为G的群G的互素同构，而P为G的α不变Sylow p-子群。假使，假设$p\notin \pi （C_G（\alpha ））$。首先，我们证明当且仅当G是p-幂幂$C_{{ñ}_G（P）}（\alpha ）$集中P。如果G是$\mathrm{小号}ž（2^{\mathrm{[R}}）$ 和 $P\mathrm{小号}\mathrm{大号}（2，2^{\mathrm{[R}}）$-在哪里免费 $\mathrm{[R}=|\alpha |$，我们证明当且仅当G是p-封闭的$C_G（\alpha ）$标准化P。由于这两个结果，我们得出$G\cong P\times H$对于H组，当且仅当$C_G（\alpha ）$集中P。我们还证明了Frobenius p-幂零定理对于接纳互素数阶自同构群的群的推广。