﻿ 有限生成无挠幂零群的有限扩张的4阶自同构 A Finite Extension of a Finitely Generated Torsion-Free Nilpotent Groups with Automorphisms of Order Four

Pure Mathematics
Vol.07 No.03(2017), Article ID:20516,4 pages
10.12677/PM.2017.73019

A Finite Extension of a Finitely Generated Torsion-Free Nilpotent Groups with Automorphisms of Order Four

Xiaodi Ma1, Yanping Zhang2, 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: Apr. 26th, 2017; accepted: May 12th, 2017; published: May 16th, 2017

ABSTRACT

Let G be a finite extension of a finitely generated torsion-free nilpotent group and be 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 an Abelian group.

Keywords:Finitely Generated, Torsion-Free Nilpotent Group, Finite Extension, Automorphism

1南京理工大学计算机科学与工程学院，江苏 南京

2河北工程大学数理学院，河北 邯郸

1. 引言和主要结果

(i)

(ii)是Abel群。

(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) 易知是满射，由引理2.3可知是正则自同构。

(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

。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

A Finite Extension of a Finitely Generated Torsion-Free Nilpotent Groups with Automorphisms of Order Four[J]. 理论数学, 2017, 07(03): 155-158. http://dx.doi.org/10.12677/PM.2017.73019

1. 1. Robinson, D.J.S. (1996) A Course in the Theory of Groups. 2nd Edition, Springer-Verlag, New York. https://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. Higman, G. (1957) Groups and Rings Having Automorphisms without Non-Trivial Fixed Elements. Journal of the London Mathematical Society, s1-32, 321-334. https://doi.org/10.1112/jlms/s1-32.3.321

4. 4. Tao, X. and Liu, H.G. (2016) Finitely Generated Torsion-Free Nilpotent Groups Admitting an Automorphism of Prime Order. Communications in Mathematical Research, 32, 167-172.

5. 5. 马晓迪, 徐涛. 有限生成无挠幂零群的4阶自同构[J]. 理论数学, 2016, 6(5): 437-440.

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

7. NOTES

*通讯作者。