﻿ 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

Pure Mathematics
Vol.06 No.05(2016), Article ID:18608,4 pages
10.12677/PM.2016.65059

Finitely Generated Torsion-Free Nilpotent Groups Admitting an Automorphism of Order Four

Xiaodi Ma1, Tao Xu2*

1College of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing Jiangsu

2College of Science, Hebei University of Engineering, Handan Hebei

Received: Sep. 3rd, 2016; accepted: Sep. 19th, 2016; published: Sep. 26th, 2016

ABSTRACT

Let G be a finitely generated torsion-free nilpotent group and a an automorphism of order four of G. If the map defined by is surjective, then the second derived subgroup is included in the centre of G and is abelian.

Keywords:Finitely Generated, Torsion-Free Nilpotent Group, Regular Automorphism, Automorphism

1南京理工大学计算机科学与工程学院，江苏 南京

2河北工程大学理学院，河北 邯郸

1. 引言和主要结果

(i)

(ii)是Abel群。

2. 定理的证明

(i)

(ii) 对于任意的正整数诱导了的正则自同构

(ii) 任取，使得。在中，我们应用引理2.1可得

(ii) 记，只需证是Abel群即可。取，考虑。如果，则的2阶正则自同构。由命题1.1知道是Abel群。因此对于任意的，有。即。因为，所以。这表明是Abel群。显然也是Abel群。如果，则-不变，因此的1阶或2阶自同构。注意到

Finitely Generated Torsion-Free Nilpotent Groups Admitting an Automorphism of Order Four[J]. 理论数学, 2016, 06(05): 437-440. http://dx.doi.org/10.12677/PM.2016.65059

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

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

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

*通讯作者。