Geodesic
On unit group of finite semisimple group algebras of non-metabelian groups up to order 72
Mathematica Bohemica, Tome 146 (2021) no. 4, pp. 429-455
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We characterize the unit group of semisimple group algebras $\mathbb {F}_qG$ of some non-metabelian groups, where $F_q$ is a field with $q=p^k$ elements for $p$ prime and a positive integer $k$. In particular, we consider all 6 non-metabelian groups of order 48, the only non-metabelian group $((C_3\times C_3)\rtimes C_3)\rtimes C_2$ of order 54, and 7 non-metabelian groups of order 72. This completes the study of unit groups of semisimple group algebras for groups upto order 72.
We characterize the unit group of semisimple group algebras $\mathbb {F}_qG$ of some non-metabelian groups, where $F_q$ is a field with $q=p^k$ elements for $p$ prime and a positive integer $k$. In particular, we consider all 6 non-metabelian groups of order 48, the only non-metabelian group $((C_3\times C_3)\rtimes C_3)\rtimes C_2$ of order 54, and 7 non-metabelian groups of order 72. This completes the study of unit groups of semisimple group algebras for groups upto order 72.
DOI : 10.21136/MB.2021.0116-19
Classification : 16U60, 20C05
Keywords: unit group; finite field; Wedderburn decomposition 
@article{10_21136_MB_2021_0116_19,
     author = {Mittal, Gaurav and Sharma, Rajendra Kumar},
     title = {On unit group of finite semisimple group algebras of non-metabelian groups up to order 72},
     journal = {Mathematica Bohemica},
     pages = {429--455},
     year = {2021},
     volume = {146},
     number = {4},
     doi = {10.21136/MB.2021.0116-19},
     mrnumber = {4336549},
     zbl = {07442512},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2021.0116-19/}
}
TY  - JOUR
AU  - Mittal, Gaurav
AU  - Sharma, Rajendra Kumar
TI  - On unit group of finite semisimple group algebras of non-metabelian groups up to order 72
JO  - Mathematica Bohemica
PY  - 2021
SP  - 429
EP  - 455
VL  - 146
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2021.0116-19/
DO  - 10.21136/MB.2021.0116-19
LA  - en
ID  - 10_21136_MB_2021_0116_19
ER  - 
%0 Journal Article
%A Mittal, Gaurav
%A Sharma, Rajendra Kumar
%T On unit group of finite semisimple group algebras of non-metabelian groups up to order 72
%J Mathematica Bohemica
%D 2021
%P 429-455
%V 146
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2021.0116-19/
%R 10.21136/MB.2021.0116-19
%G en
%F 10_21136_MB_2021_0116_19
Mittal, Gaurav; Sharma, Rajendra Kumar. On unit group of finite semisimple group algebras of non-metabelian groups up to order 72. Mathematica Bohemica, Tome 146 (2021) no. 4, pp. 429-455. doi: 10.21136/MB.2021.0116-19

[1] Bovdi, A. A., Kurdics, J.: Lie properties of the group algebra and the nilpotency class of the group of units. J. Algebra 212 (1999), 28-64. | DOI | MR | JFM

[2] Bovdi, V., Salim, M.: On the unit group of a commutative group ring. Acta Sci. Math. 80 (2014), 433-445. | DOI | MR | JFM

[3] Creedon, L., Gildea, J.: The structure of the unit group of the group algebra $F_{2^k}D_8$. Can. Math. Bull. 54 (2011), 237-243. | DOI | MR | JFM

[4] Ferraz, R. A.: Simple components of the center of $FG/J(FG)$. Commun. Algebra 36 (2008), 3191-3199. | DOI | MR | JFM

[5] Gildea, J.: The structure of the unit group of the group algebra $F_{2^k}A_4$. Czech. Math. J. 61 (2011), 531-539. | DOI | MR | JFM

[6] Gildea, J., Monaghan, F.: Units of some group algebras of groups of order 12 over any finite field of characteristic 3. Algebra Discrete Math. 11 (2011), 46-58. | MR | JFM

[7] Hurley, T.: Convolutional codes from units in matrix and group rings. Int. J. Pure Appl. Math. 50 (2009), 431-463. | MR | JFM

[8] Karpilovsky, G.: The Jacobson Radical of Group Algebras. North-Holland Mathematics Studies 135. North-Holland, Amsterdam (1987). | DOI | MR | JFM

[9] Khan, M., Sharma, R. K., Srivastava, J. B.: The unit group of $FS_4$. Acta Math. Hung. 118 (2008), 105-113. | DOI | MR | JFM

[10] Kumar, Y.: On The Unit Group Of Certain Finite Group Algebras. PhD Thesis. Indian Institute of Technology Delhi (IIT Delhi), New Delhi (2019), Available at \brokenlink{ http://{eprint.iitd.ac.in/bitstream/handle/2074/8276/TH-5966.pdf?sequence=1}}\kern0pt

[11] Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge (1994). | DOI | MR | JFM

[12] Maheshwari, S., Sharma, R. K.: The unit group of group algebra $F_qSL(2;Z_3)$. J. Algebra Comb. Discrete Struct. Appl. 3 (2016), 1-6. | DOI | MR | JFM

[13] Makhijani, N., Sharma, R. K., Srivastava, J. B.: The unit group of $F_q[D_{30}]$. Serdica Math. J. 41 (2015), 185-198. | MR

[14] Makhijani, N., Sharma, R. K., Srivastava, J. B.: A note on the structure of $F_{p^k}A_5/J(F_{p^k}A_5)$. Acta Sci. Math. 82 (2016), 29-43. | DOI | MR | JFM

[15] Makhijani, N., Sharma, R. K., Srivastava, J. B.: Units in finite dihedral and quaternion group algebras. J. Egypt. Math. Soc. 24 (2016), 5-7. | DOI | MR | JFM

[16] Pazderski, G.: The orders to which only belong metabelian groups. Math. Nachr. 95 (1980), 7-16. | DOI | MR | JFM

[17] Perlis, S., Walker, G. L.: Abelian group algebras of finite order. Trans. Am. Math. Soc. 68 (1950), 420-426. | DOI | MR | JFM

[18] Milies, C. Polcino, Sehgal, S. K.: An Introduction to Group Rings. Algebras and Applications 1. Kluwer Academic Publishers, Dordrecht (2002). | MR | JFM

[19] Sharma, R. K., Srivastava, J. B., Khan, M.: The unit group of $FA_4$. Publ. Math. 71 (2007), 21-26. | MR | JFM

[20] Sharma, R. K., Srivastava, J. B., Khan, M.: The unit group of $FS_3$. Acta Math. Acad. Paedagog. Nyházi 23 (2007), 129-142. | MR | JFM

[21] Tang, G., Wei, Y., Li, Y.: Unit groups of group algebras of some small groups. Czech. Math. J. 64 (2014), 149-157. | DOI | MR | JFM

Cité par Sources :