Group of units of finite group algebras of groups of order 24

M.  Sahai; S. F.  Ansari

doi:10.37863/umzh.v75i2.6680

M. Sahai University of Lucknow, India
S. F. Ansari University of Lucknow, India

DOI: https://doi.org/10.37863/umzh.v75i2.6680

Keywords: Group Algebras, Unit Groups

Abstract

UDC 512.5

Let $F$ be a finite field of characteristic $p.$ The structures of the unit groups of group algebras over $F$ of the three groups $D_{24},$ $S_4$ and $SL(2, \mathbb{Z}_3)$ of order $24$ are completely described in numerous works. We present the unit groups of the group algebras over $F$ for the remaining groups of order $24,$ namely, $C_{24},$ $C_{12} \times C_2,$ $C_2^3 \times C_3,$ $C_3 \rtimes C_8,$ $C_3 \rtimes Q_8,$ $D_6 \times C_4,$ $C_6 \rtimes C_4,$ $C_3 \rtimes D_8,$ $C_3 \times D_8,$ $C_3 \times Q_8,$ $A_4 \times C_2,$ and $D_{12} \times C_2.$

References

A. A. Bovdi, J. Kurdics, Lie properties of the group algebra and the nilpotency class of the group of units, J. Algebra, 212, № 1, 28–64 (1999). DOI: https://doi.org/10.1006/jabr.1998.7617

R. A. Ferraz, Simple components of the center of $FG/J(FG)$, Commun. Algebra, 36, № 9, 3191–3199 (2008). DOI: https://doi.org/10.1080/00927870802103503

The GAP Groups. GAP-Groups, Algorithms and Programming, Version 4.7.8 (2015); http://www.gap-system.org.

T. Hurley, Group rings for communications, Int. J. Group Theory, 4, 1–23 (2015).

M. Khan, R. K. Sharma, J. B. Srivastava, The unit group of $FS_4$, Acta Math. Hungar., 118, № 1-2, 105–113 (2008). DOI: https://doi.org/10.1007/s10474-007-6169-4

S. Maheshwari, The unit group of group algebras $F_qSL(2, mathbb{Z}_3)$, J. Algebra Comb. Discrete Struct. and Appl., 3, № 4, 1–6 (2016). DOI: https://doi.org/10.13069/jacodesmath.83854

S. Maheshwari, R. K. Sharma, A note on units in $F_qSL(2, mathbb{Z}_3)$, Ukr. Math. J., 73, № 8, 1331–1337 (2022); DOI: 10.1007/s11253-022-01994-7. DOI: https://doi.org/10.1007/s11253-022-01994-7

C. P. Milies, S. K. Sehgal, An introduction to group ring, Algebra and Appl., vol. 1, Kluwer Acad. Publ., Dordrecht (2002).

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, 133–161 (2012).

V. S. Pless, W. C. Huffman, Handbook of coding theory, Elsevier, New York (1998).

R. E. Sabin, S. J. Lomonaco, Metacyclic error correcting codes, Appl. Algebra Engrg. Comm. and Comput., 6, № 3, 191–210 (1995). DOI: https://doi.org/10.1007/BF01195337

M. Sahai, S. F. Ansari, Unit groups of group algebras of certain dihedral groups-II, Asian-Eur. J. Math., 12, № 4 (2018). DOI: https://doi.org/10.1142/S1793557119500669

M. Sahai, S. F. Ansari, Unit groups of finite group algebras of Abelian groups of order at most $16$, Asian-Eur. J. Math., 14, № 3 (2021). DOI: https://doi.org/10.1142/S1793557121500303

M. Sahai, S. F. Ansari, Unit groups of group algebras of groups of order 18, Comm. Algebra, 49, № 8, 3273–3282 (2021). DOI: https://doi.org/10.1080/00927872.2021.1893740

G. Tang, G. Yanyan, The unit group of $FG$ of group with order ${12}$, Int. J. Pure and Appl. Math., 73, № 2, 143–158 (2011).

G. Tang, Y. Wei, Y. Li, Unit groups of group algebras of some small groups, Czechoslovak Math. J., 64, № 1, 149–157 (2014). DOI: https://doi.org/10.1007/s10587-014-0090-0