# 有限生成无挠幂零群的4阶自同构Finitely Generated Torsion-Free Nilpotent Groups Admitting an Automorphism of Order Four

Abstract: Let G be a finitely generated torsion-free nilpotent group and α an automorphism of order four of G. If the map GG defined by is surjective, then the second derived subgroup G'' is included in the centre of G and CG(α2) is abelian.

[1] Robinson, D.J.S. (1996) A Course in the Theory of Groups. 2nd Edition, Springer-Verlag, New York. http://dx.doi.org/10.1007/978-1-4419-8594-1

[2] Burnside, W. (1955) Theory of Groups of Finite Order. 2nd Edition, Dover Publications Inc., New York.

[3] Gorenstein, D. (1980) Finite Groups. Chelsea Publishing Company, New York.

[4] Neumann, B.H. (1956) Group with Automorphisms That Leave Only the Neutral Element Fixed. Archiv der Mathematik, 7, 1-5. http://dx.doi.org/10.1007/BF01900516

[5] Thompson, J. (1959) Finite Groups with Fixed-Point-Free Automorphisms of Prime Order. Proc. Nat. Acad. Sci., 45: 578-581. http://dx.doi.org/10.1073/pnas.45.4.578

[6] Higman, G. (1957) Groups and Rings Having Automorphisms without Non-Trivial Fixed Elements. Journal of the London Mathematical Society, 64, 321-334. http://dx.doi.org/10.1112/jlms/s1-32.3.321

[7] 徐涛, 刘合国. 有限秩的可解群的正则自同构[J]. 数学年刊, 2014, 35A(5): 543-550.

[8] Xu, T. and Liu, H.G. (2016) Finitely Generated Torsion-Free Nilpotent Groups Admitting an Automorphism of Prime Order. Communications in Mathematical Sciences, 32, 167-172.

[9] Kovács, L.G. (1961) Group with Regular Automorphisms of Order Four. Mathematische Zeitschrift, 75, 277-294. http://dx.doi.org/10.1007/BF01211026

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