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 F G be a group algebra of a group G over a field F and 𝒰 ( F G ) the unit group of F G . 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 F A 4 over any finite field of characteristic 3 and the structure of the unit group of F Q 12 over any finite field of characteristic 2 , where Q 12 = 〈 x , y ; x 6 = 1 , y 2 = x 3 , x y = x - 1 〉 .

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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