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Polycyclic group admitting an almost regular automorphism of prime order. (English) Zbl 1202.20037
By J. G. Thompson’s theorem [Proc. Natl. Acad. Sci. USA 45, 578-581 (1959; Zbl 0086.25101)], if a finite group has a fixed-point-free automorphism of prime order, then the group is nilpotent. By G. Higman’s theorem [J. Lond. Math. Soc. 32, 321-334 (1957; Zbl 0079.03203)] the nilpotency class of a (locally) nilpotent group with a fixed-point-free automorphism of prime order $$p$$ is bounded above by a function $$h(p)$$ of $$p$$ only.
In the present paper it is proved that if a polycyclic group $$G$$ has an automorphism $$\varphi$$ of prime order $$p$$ with finite fixed-point subgroup (centralizer) $$C_G(\varphi)$$, then $$G$$ has a subgroup of finite index that is nilpotent of class at most $$h(p)$$.
The proof does not use Thompson’s theorem but relies on Higman’s theorem applied to certain finite nilpotent quotients of $$G$$. Examples show that $$G$$ may not be nilpotent even if $$C_G(\varphi)=1$$. It is also not possible to bound the index of a nilpotent subgroup in terms of $$p$$ and $$|C_G(\varphi)|$$. This is in contrast to the case of locally finite and locally nilpotent groups with such an automorphism, where such bounds do exist (albeit with a worse bound for the nilpotency class of a subgroup, which however still depends only on $$p$$) [P. Fong, Osaka J. Math. 13, 483-489 (1976; Zbl 0372.20010), B. Hartley, T. Meixner, Arch. Math. 36, 211-213 (1981; Zbl 0447.20014), E. I. Khukhro, Mat. Sb. 181, No. 9, 1207-1219 (1990; Zbl 0713.17013), Yu. A. Medvedev, J. Algebra 164, No. 3, 877-885 (1994; Zbl 0811.20033)].
In the concluding remarks the author considers a polycyclic group $$G$$ with an automorphism $$\varphi$$ of arbitrary finite order $$n$$ with finite centralizer $$C_G(\varphi)$$. For $$n=4$$ similar reduction to finite nilpotent quotients of $$G$$ makes it possible to use L. G. Kovács’s theorem [Math. Z. 75, 277-294 (1961; Zbl 0097.01202)], so then $$G$$ has a subgroup of finite index which is centre-by-metabelian. For arbitrary $$n$$ it is still unknown whether the derived length of a finite (nilpotent) group with a fixed-point-free coprime automorphism of order $$n$$ is bounded. However, such a result is known for Lie algebras as V. A. Kreknin’s theorem [Sov. Math., Dokl. 4, 683-685 (1963); translation from Dokl. Akad. Nauk SSSR 150, 467-469 (1963; Zbl 0134.03604], with Kreknin’s function $$k(n)$$ bounding the derived length of a Lie algebra with a fixed-point-free automorphism of order $$n$$. As pointed out to the author by the referee, Kreknin’s theorem can be used in the general case to obtain a subgroup of finite index that is soluble of derived length at most $$k(n)+1$$.

##### MSC:
 20E36 Automorphisms of infinite groups 20F19 Generalizations of solvable and nilpotent groups 20F28 Automorphism groups of groups 20F40 Associated Lie structures for groups 20E07 Subgroup theorems; subgroup growth
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##### References:
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