Unit groups of group algebras of some small groups

Gaohua Tang; Yangjiang Wei; Yuanlin Li

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 1, page 149-157
  • ISSN: 0011-4642

Abstract

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Let be a group algebra of a group over a field and the unit group of . It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group with order over any finite field of characteristic is established. We also characterize the structure of the unit group of over any finite field of characteristic and the structure of the unit group of over any finite field of characteristic , where .

How to cite

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Tang, Gaohua, Wei, Yangjiang, and Li, Yuanlin. "Unit groups of group algebras of some small groups." Czechoslovak Mathematical Journal 64.1 (2014): 149-157. <http://eudml.org/doc/262013>.

@article{Tang2014,
abstract = {Let $FG$ be a group algebra of a group $G$ over a field $F$ and $\{\mathcal \{U\}\}(FG)$ the unit group of $FG$. It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group $G$ with order $21$ over any finite field of characteristic $3$ is established. We also characterize the structure of the unit group of $FA_4$ over any finite field of characteristic $3$ and the structure of the unit group of $FQ_\{12\}$ over any finite field of characteristic $2$, where $Q_\{12\}=\langle x, y; x^6=1, y^2=x^3, x^y=x^\{-1\} \rangle $.},
author = {Tang, Gaohua, Wei, Yangjiang, Li, Yuanlin},
journal = {Czechoslovak Mathematical Journal},
keywords = {group ring; unit group; augmentation ideal; Jacobson radical; group algebras of finite groups; group rings; groups of units; augmentation ideals; Jacobson radical},
language = {eng},
number = {1},
pages = {149-157},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Unit groups of group algebras of some small groups},
url = {http://eudml.org/doc/262013},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Tang, Gaohua
AU - Wei, Yangjiang
AU - Li, Yuanlin
TI - Unit groups of group algebras of some small groups
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 1
SP - 149
EP - 157
AB - Let $FG$ be a group algebra of a group $G$ over a field $F$ and ${\mathcal {U}}(FG)$ the unit group of $FG$. It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group $G$ with order $21$ over any finite field of characteristic $3$ is established. We also characterize the structure of the unit group of $FA_4$ over any finite field of characteristic $3$ and the structure of the unit group of $FQ_{12}$ over any finite field of characteristic $2$, where $Q_{12}=\langle x, y; x^6=1, y^2=x^3, x^y=x^{-1} \rangle $.
LA - eng
KW - group ring; unit group; augmentation ideal; Jacobson radical; group algebras of finite groups; group rings; groups of units; augmentation ideals; Jacobson radical
UR - http://eudml.org/doc/262013
ER -

References

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  1. Brockhaus, P., 10.1016/0021-8693(85)90117-6, J. Algebra 95 (1985), 454-472. (1985) Zbl0568.20010MR0801281DOI10.1016/0021-8693(85)90117-6
  2. Chen, W., Xie, C., Tang, G., The unit groups of of groups with order , J. Guangxi Teachers Education University 30 (2013), 14-20. (2013) MR3162615
  3. Creedon, L., The unit group of small group algebras and the minimum counterexample to the isomorphism problem, Int. J. Pure Appl. Math. 49 (2008), 531-537. (2008) Zbl1192.16035MR2482633
  4. Creedon, L., Gildea, J., 10.4153/CMB-2010-098-5, Can. Math. Bull. 54 (2011), 237-243. (2011) MR2884238DOI10.4153/CMB-2010-098-5
  5. Creedon, L., Gildea, J., The structure of the unit group of the group algebra , Int. J. Pure Appl. Math. 45 (2008), 315-320. (2008) MR2421868
  6. Gildea, J., The structure of , Int. Electron. J. Algebra (electronic only) 8 (2010), 153-160. (2010) MR2660546
  7. Gildea, J., 10.1080/00927872.2010.482552, Commun. Algebra 38 (2010), 3311-3317. (2010) MR2724220DOI10.1080/00927872.2010.482552
  8. Gildea, J., 10.1007/s10587-011-0071-5, Czech. Math. J. 61 (2011), 531-539. (2011) MR2905421DOI10.1007/s10587-011-0071-5
  9. Gildea, J., 10.1142/S0218196710005856, Int. J. Algebra Comput. 20 (2010), 721-729. (2010) Zbl1205.16031MR2726571DOI10.1142/S0218196710005856
  10. Gildea, J., 10.1007/s00025-011-0094-0, Results Math. 61 (2012), 245-254. (2012) MR2925119DOI10.1007/s00025-011-0094-0
  11. Gildea, J., Monaghan, F., Units of some group algebras of groups of order over any finite field of characteristic , Algebra Discrete Math. 11 (2011), 46-58. (2011) Zbl1256.16023MR2868359
  12. Nezhmetdinov, T. I., 10.1080/00927870903451918, Commun. Algebra 38 (2010), 4669-4681. (2010) Zbl1216.16026MR2805136DOI10.1080/00927870903451918
  13. Passman, D. S., The Algebraic Structure of Group Rings, Pure and Applied Mathematics Wiley, New York (1977). (1977) Zbl0368.16003MR0470211
  14. Milies, C. Polcino, Sehgal, S. K., 10.1007/978-94-010-0405-3_3, Algebras and Applications 1 Kluwer Academic Publishers, Dordrecht (2002). (2002) MR1896125DOI10.1007/978-94-010-0405-3_3
  15. Sharma, R. K., Srivastava, J. B., Khan, M., The unit group of , Publ. Math. 71 (2007), 21-26. (2007) Zbl1135.16033MR2340031
  16. Tang, G., Gao, Y., The unit group of of groups with order , Int. J. Pure Appl. Math. 73 (2011), 143-158. (2011) MR2933951

Citations in EuDML Documents

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  1. Gaurav Mittal, Rajendra Kumar Sharma, On unit group of finite semisimple group algebras of non-metabelian groups up to order 72
  2. Rajendra K. Sharma, Gaurav Mittal, On the unit group of a semisimple group algebra
  3. Navamanirajan Abhilash, Elumalai Nandakumar, Rajendra K. Sharma, Gaurav Mittal, Structure of the unit group of the group algebras of non-metabelian groups of order 128

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