有限ATI-群的类保持Coleman自同构

海进科, 李正兴

数学学报 ›› 2010, Vol. 53 ›› Issue (5) : 891-896.

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数学学报 ›› 2010, Vol. 53 ›› Issue (5) : 891-896. DOI: 10.12386/A2010sxxb0099
论文

有限ATI-群的类保持Coleman自同构

    海进科, 李正兴
作者信息 +

Class-Preserving Coleman Automorphisms of Finite ATI-Groups

    Jin Ke HAI, Zheng Xing LI
Author information +
文章历史 +

摘要

G是一个有限群,对G的任意阿贝尔子群A及任意gG, 若AAg=1或A,则称G为一个ATI-群.本文证明了,对任意pπ(G),如果ATI-群G的一个p-方幂阶类保持自同构在G的任意Sylow子群上的限制等于G的某个内自同构的限制,则它必定是一个内自同构.作为该结果的一个直接推论,我们也证明了有限ATI-群G有正规化性质.

 

Abstract

A finite group G is called an ATI-group if for any abelian subgroup A of G, AAg=1 or A for all gG. In this paper, it is shown that a class-preserving automorphism of an ATI-group G of p-power order whose restriction to any Sylow subgroup of G is equal to the restriction of some inner automorphism of G is necessarily an inner automorphism. As an immediate consequence, we obtain that the normalizer property holds for finite ATI-groups.

 

关键词

自同构群 / 类保持自同构及Coleman自同构 / ATI-群

Key words

automorphism groups / class-preserving automorphisms and Coleman automorphisms / ATI-groups

引用本文

导出引用
海进科, 李正兴. 有限ATI-群的类保持Coleman自同构. 数学学报, 2010, 53(5): 891-896 https://doi.org/10.12386/A2010sxxb0099
Jin Ke HAI, Zheng Xing LI. Class-Preserving Coleman Automorphisms of Finite ATI-Groups. Acta Mathematica Sinica, Chinese Series, 2010, 53(5): 891-896 https://doi.org/10.12386/A2010sxxb0099

参考文献


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[2] Coleman D.B., On the modular group ring of p-group, Proc. Amer. Math. Soc., 1964, 15(4): 511--514.

[3] Hertweck M., Jespers E., Class-preserving Automorphisms and the normalizer property for Blackburn groups, J. Group Theory, 2009, 12(1): 157--169.

[4] Hai J. K., Wang Y. L., Class-preserving Coleman Automorphisms of the finite group with a T.I. Sylow 2-subgroup, Acta Math. Sin., Chinese Series, 2008, 51(6): 1115--1118 (in Chinese).

[5] Guo X. Y., Li S. R., Flacell P., Finite groups whose abelian subgroups are TI-subgroups, J. Algebra, 2007, 307(2): 565--569.

[6] Huppert B., Endliche Gruppen I, Berlin: Springer-Verlag, 1967.

[7] Hertweck M., Kimmerle W., Coleman Automorphisms of finite groups, Math. Z., 2002, 242(2): 203--215.

[8] Hertweck M., Local analysis of the normalizer problem, J. Pure Appl. Algebra, 2001, 163(3): 259--276.

[9] Hertweck M., Class-preserving Coleman Automorphisms of finite groups, Monatsh. Math., 2002, 136(1): 1--7.

[10] Li Y. L., Sehgal S. K., Parmenter M. M., On the normalizer property for integral group rings, Comm. Algebra, 1999, 27(9): 4217--4223.

基金

国家自然科学基金资助项目(10771132)山东省自然科学基金资助项目(Y2008A03)及山东省教育厅基金资助项目(J07YH06)

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