On the unit group of $ F_{q}[SL(2,F_{7})]$ and it's normal complement

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics and Scientific Computing, Madan Mohan Malaviya University of Technology Gorakhpur, India

Abstract

In this paper, we study the structure of the unit group of the semisimple group algebra $F_q[SL(2,F_{7})]$, where $F_q$ is a finite field of characteristic $p \neq 2,3,7$, and $SL(2,F_{7})$ denotes the special linear group consisting of $2 \times 2$ matrices over the finite field $F_{7}$ with determinant 1. The group $SL(2,F_{7})$ is a finite non-abelian group of order 336, and its group algebra over a finite field provides an interesting case for algebraic exploration, especially in the semisimple case, which occurs when the characteristic of the field does not divide the order of the group. We provide a detailed characterization of the unit group $U(F_q[SL(2,F_{7})])$ and examine whether $SL(2,F_{7})$ has a normal complement within this unit group. Our findings show that for all primes $p \neq 2,3,7 $, the group $SL(2,F_{7})$ does not admit a normal complement in $U(F_q[SL(2,F_{7})])$. This contributes to the broader understanding of the subgroup structure and normality conditions in unit groups of semisimple group algebras involving non-abelian quasi-simple groups.

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References

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International Journal of Group Theory

Articles in Press, Corrected Proof
Available Online from 20 May 2026

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History

  • Receive Date: 08 July 2025
  • Revise Date: 27 January 2026
  • Accept Date: 14 February 2026
  • Published Online: 20 May 2026

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