Revisiting the Zassenhaus conjecture on torsion units for the integral group rings of small groups

详细信息    查看全文

• 作者：ALLEN HERMAN ; GURMAIL SINGH
• 关键词：Integral group rings ; torsion units ; Zassenhaus conjectures ; Primary ; 20C05 ; Secondary ; 16S34
• 刊名：Proceedings Mathematical Sciences
• 出版年：2015
• 出版时间：May 2015
• 年：2015
• 卷：125
• 期：2
• 页码：167-172
• 全文大小：318 KB
• 参考文献：[1]Cohn J A and Livingstone D, On the structure of group algebras, I, Canad. J. Math. 17 (1965) 583-93MATH MathSciNet View Article
  [2]Hertweck M, Torsion units of integral group rings of certain metabelian groups, Proc. Edinburgh Math. Soc. 51 (2008) 363-85MATH MathSciNet View Article
  [3]Hertweck M, The orders of torsion units in integral group rings of finite solvable groups, Comm. Algebra 36 (10) (2008) 3585-588MATH MathSciNet View Article
  [4]H?fert C and Kimmerle W, On torsion units of integral group rings of groups of small order, in: Groups, rings and group rings, Lecture Notes in Pure and Applied Mathematics (2006) (CRC Press /Chapman and Hall) vol. 286, pp. 243-52
  [5]Luthar I S and Passi I B S, Zassenhaus conjecture for A 5, Proc. Indian Acad. Sci. (Math. Sci.) 99 (1) (1989) 1-MATH MathSciNet View Article
  [6]Sehgal S K, Units in integral group rings, Pitman Monograph and Surveys in Pure and Applied Mathematics (1993) (Harlow: Longman Scientific and Techical) vol. 69
  [7]The GAP Group, GAP -Groups, Algorithms and Programming, Version 4.6.2, Aachen, St. Andrews, 1999, ( http://?www.?gap-system.?org )
  
• 作者单位：ALLEN HERMAN (1)
  GURMAIL SINGH (1)
  
  1. Department of Mathematics and Statistics, University of Regina, Regina, S4S 0A2, Canada
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer India
• ISSN：0973-7685

文摘

In recent years several new restrictions on integral partial augmentations for torsion units of \(\mathbb {Z} G\) have been introduced, which have improved the effectiveness of the Luthar–Passi method for checking the Zassenhaus conjecture for specific groups G. In this note, we report that the Luthar–Passi method with the new restrictions are sufficient to verify the Zassenhaus conjecture with a computer for all groups of order less than 96, except for one group of order 48 -the non-split covering group of S 4, and one of order 72 of isomorphism type (C 3 × C 3) ?D 8. To verify the Zassenhaus conjecture for this group we give a new construction of normalized torsion units of \(\mathbb {Q} G\) that are not conjugate to elements of \(\mathbb {Z} G\).