# 有限生成剩余有限群的素数阶自同构Finitely Generated Residually Finite Groups with Automorphisms of Prime Order

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Let be an automorphism of prime order p of a finitely generated residually finite group G. If the map defined by is surjective, then G is nilpotent of class at most h(p), where h(p) is a function depending only on p. In particular, if is of order 2, then G is abelian.

1. 引言

Burnside [2] 证明了：具有2阶无不动点自同构的有限群是奇阶交换群。这是有限群中的一个经典的结论。随后，Neumann [3] 考虑了任意群的3阶无不动点自同构，得到了下面的结果。

2. 定理的证明及推论

$\underset{m}{\cap }{G}^{m}\le \underset{g}{\cap }{N}_{g}=1$

$\left[{g}_{1},{g}_{2},\cdots ,{g}_{h\left(p\right)+1}\right]\in {G}^{m}$

$\left[{g}_{1},{g}_{2},\cdots ,{g}_{h\left(p\right)+1}\right]\in \underset{m}{\cap }{G}^{m}=1$

$\left[{h}_{1},{h}_{2}\right]\in {N}_{g}$

$\left[{h}_{1},{h}_{2}\right]\in \underset{g}{\cap }{N}_{g}=1$

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