On the structure of the augmentation quotient group for some nonabelian 2-groups. (English) Zbl 1249.20004
Summary: Let \(G\) be a finite nonabelian group, \(\mathbb ZG\) its associated integral group ring, and \(\Delta(G)\) its augmentation ideal. For the semidihedral group and another nonabelian 2-group the problem of their augmentation ideals and quotient groups \(Q_n(G)=\Delta^n(G)/\Delta^{n+1}(G)\) is dealt with. An explicit basis for the augmentation ideal is obtained, so that the structure of its quotient groups can be determined.
MSC:
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
| 16S34 | Group rings |
Keywords:
integral group rings; augmentation ideals; augmentation quotients; finite 2-groups; semidihedral groupsReferences:
| [1] | F. Bachman, L. Grünenfelder: The periodicity in the graded ring associated with an integral group ring. J. Pure Appl. Algebra 5 (1974), 253–264. · Zbl 0301.16011 · doi:10.1016/0022-4049(74)90036-X |
| [2] | A. Bak, G. P. Tang: Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion abelian groups. Adv. Math. 189 (2004), 1–37. · Zbl 1068.16032 · doi:10.1016/j.aim.2003.11.002 |
| [3] | D. Gorenstein: Finite Groups, 2nd ed. New York: Chelsea Publishing Company, 1980. · Zbl 0463.20012 |
| [4] | A. W. Hales, I. B. S. Passi: The augmentation quotients of finite Abelian p-groups. Contemp. Math. 93 (1989), 167–171. · Zbl 0677.20006 · doi:10.1090/conm/093/1003351 |
| [5] | M. M. Parmenter: A basis for powers of the augmentation ideal. Algebra Colloq. 8 (2001), 121–128. · Zbl 0979.16015 |
| [6] | I. B. S. Passi: Group Rings and Their Augmentation Ideals. Lecture Notes in Mathematics. 715, Springer-Verlag, Berlin, 1979. · Zbl 0405.20007 |
| [7] | G. P. Tang: On a problem of Karpilovsky. Algebra Colloq. 10 (2003), 11–16. · Zbl 1034.20006 · doi:10.1007/s100110300002 |
| [8] | H. Zhao, G. Tang: Structure of powers of augmentation ideals and their quotient groups for integral group rings of dihedral groups. J. Shaanxi Norm. Univ., Nat. Sci. Ed. 33 (2005), 18–21. (In Chinese.) · Zbl 1084.20003 |
| [9] | Q. Zhou, H. You: Augmentation quotients of the dihedral group. Chin. Ann. Math., Ser. A 31 (2010), 531–540. (In Chinese.) · Zbl 1224.20001 |
| [10] | Q. Zhou, H. You: On the structure of augmentation quotient groups for generalized quaternion group. Algebra Colloq. To appear. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
