The Structure of the Unit Group of $\mathbb{F}_{3^{k}}(C_{3}\times D_{2n})$ and Finite Group Algebras of Groups of Order $42$

Abstract

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p>0$ with $|\mathbb{F}_{q}|=q=p^{k}$ and $\mathcal{U}(\mathbb{F}_{q}G)$ be the unit group of the group algebra $\mathbb{F}_{q}G$ for some group $G$. There are $6$ groups of order $42$ up to isomorphism. In this paper, we provide a characterization of $\mathcal{U}(\mathbb{F}_{3^{k}}(C_{3}\times D_{2n}))$ and establish the structures of the unit groups of some finite group algebras of groups of order $42$.

Keywords

• Group algebra
• Jacobson radical
• Wedderburn decomposition
• unit group

References

1. S. F. Ansari and M. Sahai, Unit groups of group algebras of groups of order $20$, Quaest. Math., 44(4) (2021), 503-511.
2. A. Bovdi and L. Erdei, Unitary units in modular group algebras of $2$-groups, Comm. Algebra, 28(2) (2000), 625-630.
3. V. A. Bovdi and A. N. Grishkov, Unitary and symmetric units of a commutative group algebra, Proc. Edinb. Math. Soc. (2), 62(3) (2019), 641-654.
4. L. Creedon and J. Gildea, The structure of the unit group of the group algebra $\mathbb{F}_{3^{k}}D_{6}$, Int. J. Pure Appl. Math., 45(2) (2008), 315-320.
5. L. Creedon and J. Gildea, The structure of the unit group of the group algebra $\mathbb{F}_{2^{k}}D_{8}$, Canad. Math. Bull., 54(2) (2011), 237-243.
6. R. A. Ferraz, Simple components of the center of $FG/J(FG)$, Comm. Algebra, 36(9) (2008), 3191-3199.
7. J. Gildea and R. Taylor, Units of the group algebra of the group $C_{n}\times D_{6}$ over any finite field of characteristic $3$, Int. Electron. J. Algebra, 24 (2018), 62-67.
8. K. Kaur and M. Khan, Units in $F_{2}D_{2p}$, J. Algebra Appl., 13(2) (2014), 1350090 (9 pp).

9. N. Makhijani, R. K. Sharma and J. B. Srivastava, A note on units in $\mathbb{F}_{p^m}D_{2p^n}$, Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 30(1) (2014), 17-25.
10. N. Makhijani, R. K. Sharma and J. B. Srivastava, Units in $\mathbb{F}_{2^{k}}D_{2n}$, Int. J. Group Theory, 3(3) (2014), 25-34.
11. N. Makhijani, R. K. Sharma and J. B. Srivastava, The unit group of $\mathbb{F}_{q}[D_{30}]$, Serdica Math. J., 41(2-3) (2015), 185-198.
12. S. Malik and R. K. Sharma, Describing the group of units of integral grouprings $\mathbb{Z}D_{8}$ and $\mathbb{Z}D_{12}$, Asian-Eur. J. Math., 17(1) (2024), 2350236 (13 pp).
13. S. Malik, R. K. Sharma and M. Sahai, The structure of the unit group of the group algebras $\mathbb{F}_{3^{k}}D_{6n}$ and $\mathbb{F}_{q}D_{42}$, Ann. Math. Inform., DOI: 10.33039/ami.2024.08.001.
14. G. Mittal and R. K. Sharma, Wedderburn decomposition of a semisimple group algebra $\mathbb{F}_{q}G$ from a subalgebra of factor group of $G$, Int. Electron. J. Algebra, 32 (2022), 91-100.
15. D. S. Passman, The Algebraic Structure of Group Rings, Wiley Interscience, New York, 1977.
16. C. Polcino Milies and S. K. Sehgal, An Introduction to Group Rings, Kluwer Academic Publishers, Dordrecht, 2002.
17. M. Sahai and S. F. Ansari, Unit groups of group algebras of certain dihedral groups-II, Asian-Eur. J. Math., 12(4) (2019), 1950066 (12 pp).
18. M. Sahai and S. F. Ansari, Unit groups of finite group algebras of abelian groups of order at most $16$, Asian-Eur. J. Math., 14(3) (2021), 2150030 (17 pp).
19. M. Sahai and S. F. Ansari, Unit groups of group algebras of groups of order $18$, Comm. Algebra, 49(8) (2021), 3273-3282.
20. R. K. Sharma, J. B. Srivastava and M. Khan, The unit group of $FS_{3}$, Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 23(2) (2007), 129-142.
21. G. Tang, Y. Wei and Y. Li, Unit groups of group algebras of some small groups, Czechoslovak Math. J., 64(139)(1) (2014), 149-157.