The unit group of the group algebra $\mathbb{F}_{q}(\mathbb{Z}_9\rtimes\mathbb{Z}_3)$

R. K. Sharma; Yogesh Kumar; D. C. Mishra

doi:10.3842/umzh.v76i7.7500

R. K. Sharma Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India
Yogesh Kumar Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India
D. C. Mishra Department of Mathematics, Govt. P. G. College Jaiharikhal, Uttrakhand, India

DOI: https://doi.org/10.3842/umzh.v76i7.7500

Анотація

УДК 512.5

Група одиниць групової алгебри $\mathbb{F}_{q}(\mathbb{Z}_9\rtimes\mathbb{Z}_3)$

Нехай $p$ – просте число, $\mathbb{F}_q$ – скінченне поле з $q=p^n$ елементами, а $\mathbb{Z}_9\rtimes\mathbb{Z}_3$ --- напівпрямий добуток груп $\mathbb{Z}_9 $ і $\mathbb{Z}_3$. Наведено повну характеризацію групи одиниць $\mathcal{U}(\mathbb{F}_q(\mathbb{Z}_9\rtimes\mathbb{Z}_3))$ групової алгебри $\mathbb{F}_q(\mathbb{ Z}_9\rtimes\mathbb{Z}_3).$

Посилання

A. A. Bovdi, Group rings} (in English), Uzhgorod (1974).

C. W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras, Pure and Appl. Math., vol.~11, John Wiley and Sons, New York, London (1962).

A. Dholakia, Introduction to convolutional codes and applications, Kluwer Acad. Publ., Norwell, USA (1994). DOI: https://doi.org/10.1007/978-1-4615-2712-1

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

J. Gildea, F. Monaghan, Unit of some group algebras of group of order $12$ over any field of characteristic $3$, Algebra and Discrete Math., 11, № 1, 46–58 (2011).

J. Gildea, The structure of the unitary units of the group algebra $F_{2^{k}}D_{8}$, Int. Electron. J. Algebra, 9, 171–176 (2011).

J. Gildea, The structure of the unit group of the group algebra $F_{3^{k}} (C_{3} times D_{6})$, Comm. Algebra, 38, № 9, 3311–331 (2010). DOI: https://doi.org/10.1080/00927872.2010.482552

J. Gildea, The structure of the unit group of the group algebra $F_{2^{k}}A_{4}$, Czechoslovak Math. J., 61, № 2, 531–539 (2011). DOI: https://doi.org/10.1007/s10587-011-0071-5

J. Gildea, Units of the group algebra $F_{2^{k}} (C_{2}× D_{8})$, J. Algebra and Appl., 10, № 4, 643–647 (2011). DOI: https://doi.org/10.1142/S0219498811004720

J. Gildea, Units of the group algebra $F_{3^{k}} (C_{3}^{2}⋊ C_{2})$, Acta Sci. Math. (Szeged), 77, 53–62 (2011). DOI: https://doi.org/10.1007/BF03651365

J. Gildea, L. Creedon, The structure of the unit group of the group algebra $F_{2^{k}}D_{8}$, Canad. Math. Bull., 54, № 2, 237–243 (2011). DOI: https://doi.org/10.4153/CMB-2010-098-5

T. Hueley, Group rings and ring of matrices, Int. J. Pure and Appl. Math., 31, № 3, 319–335 (2006).

T. Hueley, Convolutional codes from units in matrix and group rings, Int. J. Pure and Appl. Math., 50, № 3, 431–463 (2009).

Y. Kumar, R. K. Sharma, J. B. Srivastava, Unit group of the group algebra $FS_5$, Acta Math. Acad. Paedagog. Nyhazi., 33, 187–193 (2017).

P. Kumari, M. Sahai, R. K. Sharma, Jordan regular units in rings and group rings, Ukr. Math. J., 75, № 3, 351–63 (2023). DOI: https://doi.org/10.37863/umzh.v75i3.1130

G. Karpilovsky, The Jacobson radical of group algebras, North-Holland, Amsterdam (1987).

G. Karpilovsky, Structure of blocks of group algebras, Pitman Monographs and Surveys in Pure and Appl. Math., 33, Longman Sci. and Technical, Harlow; John Wiley and Sons, Inc., New York (1987).

G. Karpilovsky, Unit groups of classical rings, Oxford Sci. Publ., The Clarendon Press, Oxford Univ. Press, New York (1988).

G. Karpilovsky, Unit groups of group rings, Pitman Monographs and Surveys in Pure and Appl. Math., 47, Longman Sci. and Technical, Harlow, John Wiley and Sons, Inc., New York (1989).

N. Makhijani, R. K. Sharma, J. B. Srivastava, A note on units in $F_{p^m}D_{2p^n}$, Acta Math. Acad. Paedagog. Nyhazi., 30}, № 1, 17–25 (2014).

N. Makhijani, R. K. Sharma, J. B. Srivastava, The unit group of algebra of circulant matrices, Int. J. Group Theory, 3, № 4, 13–16 (2014).

N. Makhijani, R. K. Sharma, J. B. Srivastava, The unit group of finite group algebra of a generalized dihedral group, Asian-Eur. J. Math., 7, 1450034 (2014); DOI:10.1142/S179355711450034X. DOI: https://doi.org/10.1142/S179355711450034X

N. Makhijani, R. K. Sharma, J. B. Srivastava, Units in $F_{2^{k}}D_{2n}$, Int. J. Group Theory, 3, № 3, 25–34 (2014).

N. Makhijani, R. K. Sharma, J. B. Srivastava, A note on the structure of/ $F_{p^{k}}A_{5}/J (F_{p^{k}}A_{5})$, Acta Sci. Math. (Szeged), 82, 29–43 (2016). DOI: https://doi.org/10.14232/actasm-014-311-2

S. Maheshwari, R. K. Sharma, A note on units in $F_qSL (2, 3)$, Ukr. Math. J., 73, № 8, 1331–1337 (2022). DOI: https://doi.org/10.1007/s11253-022-01994-7

R. K. Sharma, P. Yadav, Unit group of algebra of circulant matrices, Int. J. Group Theory, 2, № 4, 1–6 (2013).

R. K. Sharma, J. B. Srivastava, M. Khan, The unit group of/ $FS_3$, Acta Math. Acad. Paedagog. Nyhazi., 23, № 2, 129–142 (2007).

R. K. Sharma, J. B. Srivastava, M. Khan, The unit group of $FA_4$, Publ. Math. Debrecen, 71, 21–26 (2007). DOI: https://doi.org/10.5486/PMD.2007.3541