Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$

Year 2023, Volume: 34 Issue: 34, 182 - 196, 10.07.2023

Sheere Farhat Ansarı Meena Sahaı

https://doi.org/10.24330/ieja.1299278

Cited By: 1

Abstract

Let $C_n$, $Q_n$ and $D_n$ be the cyclic group, the quaternion group and the dihedral
group of order $n$, respectively. Recently, the structures of the unit groups of the finite group algebras of $2$-groups that contain a normal cyclic subgroup of index $2$ have been studied. The dihedral groups $D_{2n}, n\geq 3$ and the generalized quaternion groups $Q_{4n}, n\geq 2$ also contain a normal cyclic subgroup of index $2$. The structures of the unit groups of the finite group algebras $FQ_{12}$, $FD_{12}$, $F(C_2 \times Q_{12})$ and $F(C_2 \times D_{12})$ over a finite field $F$ are well known. In this article, we continue this investigation and establish the structures of the unit groups of the group algebras $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$ over a finite field $F$ of characteristic $p$ containing $p^k$ elements.

Keywords

Group algebra, unit group, dihedral group, cyclic group, quaternion group

References

• Zs. Balogh and A. Bovdi, On units of group algebras of $2$-groups of maximal class, Comm. Algebra, 32(8) (2004), 3227-3245.
• Zs. Balogh, The structure of the unit group of some group algebras, Miskolc Math. Notes, {21}(2) (2020), 615-620.
• A. Bovdi, The group of units of a group algebra of characteristic $p$, Publ. Math. Debrecen, {52}(1-2) (1998), 193-244.
• W. Burnside, Theory of Groups of Finite Order, 2nd ed., Dover Publication, Inc., New York, 1955.
• W. D. Gao, A. Geroldinger and F. Halter-Koch, Group algebras of finite abelian groups and their applications to combinatorial problems, Rocky Mountain J. Math., 39(3) (2009), 805-823.
• J. Gildea, On the order of $U(F_{p^{k}}D_{{2p^{m}}})$, Int. J. Pure Appl. Math., {46}(2) (2008), 267-272.
• J. Gildea, The centre of the maximal p-subgroup of $U(F_{p^k}D_{2p^m})$, Glasg. Math. J., {51}(3) (2009), 651-657.
• J. Gildea, The structure of the unit group of the group algebra of $F_{2^k}A_4$, Czechoslovak Math. J., {61}(136) (2011), 531-539.
• J. Gildea and F. Monaghan, Units of some group algebras of groups of order $12$ over any finite field of characteristic $3$, Algebra Discrete Math., {11}(1) (2011), 46-58.
• P. Hurley and T. Hurley, Codes from zero-divisors and units in group rings, Int. J. Inf. Coding Theory, {1}(1) (2009), 57-87.
• K. Kaur and M. Khan, Units in $F_2D_{2p}$, J. Algebra Appl., {13}(2) (2014), 1350090 (9 pp).
• S. Maheshwari, Finitely Presented Groups and Units in Group Ring, Ph.D. Thesis, IIT Delhi, (2016).
• N. Makhijani, R. K. Sharma and J. B. Srivastava, Units in $F_{2^k}D_{2n}$, Int. J. Group Theory, {3}(3) (2014), 25-34.
• N. Makhijani, R. K. Sharma and J. B. Srivastava, A note on units in $F_{p^m}D_{{2p^{n}}}$, Acta Math. Acad. Paedagog. Nyh$ \acute{a}$zi., {30}(1) (2014), 17-25.
• N. Makhijani, R. K. Sharma and J. B. Srivastava, The unit group of algebra of circulant matrices, Int. J. Group Theory, {3}(4) (2014), 13-16.
• N. Makhijani, R. K. Sharma and J. B. Srivastava, Units in finite dihedral and quaternion group algebras, J. Egyptian Math. Soc., {24}(1) (2016), 5-7.
• N. Makhijani, R. K. Sharma and J. B. Srivastava, The unit group of some special semi-simple group algebras, Quaest. Math., {39}(1) (2016), 9-28.
• F. Monaghan, Units of some group algebras of non-abelian groups of order $24$ over any finite field of characteristic $3$, Int. Electron. J. Algebra, {12} (2012), 133-161.
• D. S. Passman, The Algebraic Structure of Group Rings, Wiley Interscience, New York, 1977.
• C. Polcino Milies and S. K. Sehgal, An Introduction to Group Rings, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002.
• M. Sahai and S. F. Ansari, Unit groups of the finite group algebras of generalized quaternion groups, J. Algebra Appl., {19}(6) (2020), 2050112 (5 pp).
• 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).
• M. Sahai and S. F. Ansari, Group of units of finite group algebras of groups of order $24$, Ukrainian Math. J., {75}(2) (2023), 215-229.
• R. K. Sharma, J. B. Srivastava and M. Khan, The unit group of $FA_4$, Publ. Math. Debrecen, {71}(1-2) (2007), 21-26.
• G. Tang and Y. Gao, The unit groups of $FG$ of groups with order ${12}$, Int. J. Pure Appl. Math., {73}(2) (2011), 143-158.
• G. Tang, Y. Wei and Y. Li, Unit groups of group algebras of some small groups, Czechoslovak Math. J., 64(139) (2014), 149-157.