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

Sheere Farhat Ansarı Meena Sahaı

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

Abstract

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.

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

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.
There are 26 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Sheere Farhat Ansarı This is me

Meena Sahaı This is me

Early Pub Date May 24, 2023
Publication Date July 10, 2023
Published in Issue Year 2023 Volume: 34 Issue: 34

Cite

APA Ansarı, S. F., & Sahaı, M. (2023). Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$. International Electronic Journal of Algebra, 34(34), 182-196. https://doi.org/10.24330/ieja.1299278
AMA Ansarı SF, Sahaı M. Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$. IEJA. July 2023;34(34):182-196. doi:10.24330/ieja.1299278
Chicago Ansarı, Sheere Farhat, and Meena Sahaı. “Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$”. International Electronic Journal of Algebra 34, no. 34 (July 2023): 182-96. https://doi.org/10.24330/ieja.1299278.
EndNote Ansarı SF, Sahaı M (July 1, 2023) Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$. International Electronic Journal of Algebra 34 34 182–196.
IEEE S. F. Ansarı and M. Sahaı, “Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$”, IEJA, vol. 34, no. 34, pp. 182–196, 2023, doi: 10.24330/ieja.1299278.
ISNAD Ansarı, Sheere Farhat - Sahaı, Meena. “Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$”. International Electronic Journal of Algebra 34/34 (July 2023), 182-196. https://doi.org/10.24330/ieja.1299278.
JAMA Ansarı SF, Sahaı M. Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$. IEJA. 2023;34:182–196.
MLA Ansarı, Sheere Farhat and Meena Sahaı. “Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$”. International Electronic Journal of Algebra, vol. 34, no. 34, 2023, pp. 182-96, doi:10.24330/ieja.1299278.
Vancouver Ansarı SF, Sahaı M. Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$. IEJA. 2023;34(34):182-96.