无限亚循环群的自同构群

刘合国, 张继平,廖军

数学学报 ›› 2009, Vol. 52 ›› Issue (6) : 1047-1054.

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数学学报 ›› 2009, Vol. 52 ›› Issue (6) : 1047-1054. DOI: 10.12386/A2009sxxb0132
论文

无限亚循环群的自同构群

    刘合国1, 张继平2,廖 军1
作者信息 +

Automorphism Groups of Infinite Metacyclic Groups

    He Guo LIU1, Ji Ping ZHANG2, Jun LIAO1
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摘要

研究了无限亚循环群的自同构, 得到了它们的自同构群, 并且证明了下面的 定理:无限亚循环群G的自同构都是内自同构当且仅当G同构于下列群之一: (i) Z∝ξ Zpl, 其中p为奇素数, l为正整数,  ξ把Z的生成元1变为ξ(1)=m时, p m, 并且Aut Zpl=Zpl*=Zp(l-1)(p-1)=<m>; (ii)无限二面体群D∞.

 

Abstract

We discuss automorphisms of infinite metacyclic groups and present their automorphism groups, and then we prove the following result: All automorphisms of an infinite metacyclic group G are inner automorphisms if and only if G is isomorphism to one of following groups: (i) Z∝ξ Zpl. where p is an odd prime, l is an integer, ξ maps the generator 1 of Z to ξ(1) = m, p m, and Zpl=Zpl*=Zp(l-1)(p-1)=<m>; (ii) The infinite dihedral group D∞.

 

关键词

无限亚循环群 / 自同构群 / 群扩张

Key words

infinite metacyclic groups / automorphism groups / Outer automorphism groups / Stable automorphisms / group extensions

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刘合国, 张继平,廖军. 无限亚循环群的自同构群. 数学学报, 2009, 52(6): 1047-1054 https://doi.org/10.12386/A2009sxxb0132
He Guo LIU, Ji Ping ZHANG, Jun LIAO. Automorphism Groups of Infinite Metacyclic Groups. Acta Mathematica Sinica, Chinese Series, 2009, 52(6): 1047-1054 https://doi.org/10.12386/A2009sxxb0132

参考文献


[1] Robinson D. J. S., A Course in the Theory of Groups (Second Edition), New York: Springer-Verlag, 1996.

[2] Fan Y., Huang P., The stable automorphism Groups of splitting extension of finite groups, Chinese Ann. Math., 2001, 22(A): 791--796 (in Chinese).

[3] Sim H., Metacyclic groups of odd order, Proc. London Math. Soc., 1994, 69(3): 42--71.

[4] Hempel C. E., Metacyclic groups, Comm. Algebra, 2000, 28(8): 3865--3897.

[5] Bidwellj N. S., Curran M. J., The automorphism group of a split metacyclic p-group, Arch. Math., 2006, 87: 488--497.

[6] Curran M. J., The automorphism group of a split metacyclic 2-group, Arch. Math., 2007, 89(1): 10--23.

基金

国家重点基础研究发展计划(2006CB805904); 国家自然科学基金资助项目(10631010,10671058); 教育部博士点基金和湖北省教育厅重大项目

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