On the structure of the augmentation quotient group for some nonabelian 2-groups

Jizhu Nan; Huifang Zhao

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 1, page 279-292
  • ISSN: 0011-4642

Abstract

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Let G be a finite nonabelian group, ℤ G its associated integral group ring, and ▵ ( 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 ) = ▵ n ( G ) / ▵ n + 1 ( G ) is deal with. An explicit basis for the augmentation ideal is obtained, so that the structure of its quotient groups can be determined.

How to cite

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Nan, Jizhu, and Zhao, Huifang. "On the structure of the augmentation quotient group for some nonabelian 2-groups." Czechoslovak Mathematical Journal 62.1 (2012): 279-292. <http://eudml.org/doc/246606>.

@article{Nan2012,
abstract = {Let $G$ be a finite nonabelian group, $\{\mathbb \{Z\}\}G$ its associated integral group ring, and $\triangle (G)$ its augmentation ideal. For the semidihedral group and another nonabelian 2-group the problem of their augmentation ideals and quotient groups $Q_\{n\}(G)=\triangle ^\{n\}(G)/\triangle ^\{n+1\}(G)$ is deal with. An explicit basis for the augmentation ideal is obtained, so that the structure of its quotient groups can be determined.},
author = {Nan, Jizhu, Zhao, Huifang},
journal = {Czechoslovak Mathematical Journal},
keywords = {integral group ring; augmentation ideal; augmentation quotient groups; finite 2-group; semidihedral group; integral group rings; augmentation ideals; augmentation quotients; finite 2-groups; semidihedral groups},
language = {eng},
number = {1},
pages = {279-292},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the structure of the augmentation quotient group for some nonabelian 2-groups},
url = {http://eudml.org/doc/246606},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Nan, Jizhu
AU - Zhao, Huifang
TI - On the structure of the augmentation quotient group for some nonabelian 2-groups
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 1
SP - 279
EP - 292
AB - Let $G$ be a finite nonabelian group, ${\mathbb {Z}}G$ its associated integral group ring, and $\triangle (G)$ its augmentation ideal. For the semidihedral group and another nonabelian 2-group the problem of their augmentation ideals and quotient groups $Q_{n}(G)=\triangle ^{n}(G)/\triangle ^{n+1}(G)$ is deal with. An explicit basis for the augmentation ideal is obtained, so that the structure of its quotient groups can be determined.
LA - eng
KW - integral group ring; augmentation ideal; augmentation quotient groups; finite 2-group; semidihedral group; integral group rings; augmentation ideals; augmentation quotients; finite 2-groups; semidihedral groups
UR - http://eudml.org/doc/246606
ER -

References

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  5. Parmenter, M. M., A basis for powers of the augmentation ideal, Algebra Colloq. 8 (2001), 121-128. (2001) Zbl0979.16015MR1838512
  6. Passi, I. B. S., Group Rings and Their Augmentation Ideals, Lecture Notes in Mathematics. 715, Springer-Verlag, Berlin (1979). (1979) Zbl0405.20007MR0537126
  7. Tang, G. P., 10.1007/s100110300002, Algebra Colloq. 10 (2003), 11-16. (2003) Zbl1034.20006MR1961501DOI10.1007/s100110300002
  8. Zhao, H., Tang, G., Structure of powers of augmentation ideals and their quotient groups for integral group rings of dihedral groups, Chinese J. Shaanxi Norm. Univ., Nat. Sci. Ed. 33 (2005), 18-21. (2005) Zbl1084.20003MR2146744
  9. Zhou, Q., You, H., Augmentation quotients of the dihedral group, Chinese Chin. Ann. Math., Ser. A 31 (2010), 531-540. (2010) Zbl1224.20001MR2760767
  10. Zhou, Q., You, H., On the structure of augmentation quotient groups for generalized quaternion group, Algebra Colloq (to appear). 

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