Wedderburn decomposition of a semisimple group algebra $\mathbb{F}_qG$ from a subalgebra of factor group of $G$

Year 2022, Volume: 32 Issue: 32, 91 - 100, 16.07.2022

Gaurav Mıttal Rajendra Kumar Sharma

https://doi.org/10.24330/ieja.1077582

Cited By: 2

Abstract

In this paper, we derive a condition under which the Wedderburn decomposition of a semisimple group algebra $\mathbb{F}_qG$ can be deduced from the Wedderburn decomposition of $\mathbb{F}_q(G/H)$, where $H$ is a normal subgroup of $G$ having two elements and $q=p^k$ for some prime $p$ and $k\in \mathbb{Z}^+$. In order to complement the abstract theory of the paper, we deduce the Wedderburn decomposition and hence the unit group of semisimple group algebra $\mathbb{F}_q(A_5\rtimes C_4)$, where $A_5\rtimes C_4$ is a non-metabelian group and $C_4$ is a cyclic group of order $4$.

Keywords

Wedderburn decomposition, unit group, finite field

References

• G. K. Bakshi, S. Gupta and I. B. S. Passi, The algebraic structure of finite metabelian group algebras, Comm. Algebra, 43(6) (2015), 2240-2257.
• R. A. Ferraz, Simple components of the center of $\mathbb{F}G/J(\mathbb{F}G)$, Comm. Algebra, 36(9) (2008), 3191-3199.
• M. Khan, R. K. Sharma and J. B. Srivastava, The unit group of $FS_4$, Acta Math. Hungar., 118(1-2) (2008), 105-113.
• R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, New York, 1994.
• N. Makhijani, R. K. Sharma and J. B. Srivastava, The unit group of $\mathbb{F}_q[D_{30}]$, Serdica Math. J., 41 (2015), 185-198.
• N. Makhijani, R. K. Sharma and J. B. Srivastava, A note on the structure of $\mathbb{F}_{p^k}A_5/J(\mathbb{F}_{p^k}A_5)$, Acta Sci. Math. (Szeged), 82 (2016), 29-43.
• C. P. Milies and S. K. Sehgal, An Introduction to Group Rings, Kluwer Academic Publishers, Dordrecht, 2002.
• G. Mittal and R. K. Sharma, On unit group of finite group algebras of non-metabelian groups upto order 72, Math. Bohem., 146(4) (2021), 429-455.
• G. Mittal and R. K. Sharma, On unit group of finite group algebras of non-metabelian groups of order 108, J. Algebra Comb. Discrete Appl., 8(2) (2021), 59-71.
• G. Mittal and R. K. Sharma, Unit group of semisimple group algebras of some non-metabelian groups of order 120, Asian-Eur. J. Math, (2021), Online first, DOI: 10.1142/S1793557122500590.
• S. Perlis and G. L. Walker, Abelian group algebras of finite order, Trans. Amer. Math. Soc., 68 (1950), 420-426.
• B. Sagan, The Symmetric Group, Representations, Combinatorial Algorithms, and Symmetric Functions, Springer-Verlag, 2001.
• R. K. Sharma and G. Mittal, Unit group of semisimple group algebra $F_qSL(2, 5)$, Math. Bohem., (2021), Online first, DOI: 10.21136/MB.2021.0104-20.
• R. K. Sharma, J. B. Srivastava and M. Khan, The unit group of FA_4, Publ. Math. Debrecen, 71(1-2) (2007), 21-26.
• P. Webb, A Course in Finite Group Representation Theory, Cambridge Studies in Advanced Mathematics, 161, Cambridge University Press, Cambridge, 2016.