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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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