Modular group algebras whose group of unitary units is locally nilpotent

Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$

“…There are 3 choices for n i 's given by (6, 6, 6, 7, 7, 8), (6, 6, 6, 8, 8, 7) and (6,7,7,8,8,6). To deduce the unique choice, we explicitly take p = 5 and k = 1.…”

“…To deduce the unique choice, we explicitly take p = 5 and k = 1. Further, we note that the group N is isomorphic to the group generated by permutations a, b , where a = (3, 8, 7, 6, 5, 4) and b = (1, 2, 6) (3,4,8). Next, we consider the map Ψ : N → GL (7,5) given as follows:…”

“…where A(g), B(g) and C(g) are square matrices with entries from F 5 depending on g. Therefore, Proposition 2.4 implies that M(7, F 5 ) must appear in WD of F 5 N . Thus, we are remaining with only two possibilities of n i 's is given by (6, 6, 6, 7, 7, 8) and (6,7,7,8,8,6). For uniqueness, we again consider a map from N → GL(8, 5) given as follows:…”

“…It has been extensively investigated how the unit group of the group algebra F q G is structured (see [1,3,4,[16][17][18]20,21,23,25,27,28]). Furthermore, there have been significant developments in the exploration of the unit group of modular group algebras, in addition to integral and semisimple group algebras (see [5][6][7][8] and the references therein for a comprehensive and recent literature in this direction). One of the most important studies in this area examines the unit groups of the semisimple group algebras of all metabelian groups (see [4]).…”

“…Further, Sharma et al [15,24] identified the unit group of the semisimple group algebra for the special linear groups SL (2,3) and SL (2,5). Sivaranjani et al [27] studied the unit group of the semisimple group algebra for the special linear groups SL (2,8) and SL (2,9). The unit group of the semisimple group algebra for the general linear group GL(2, 3) is studied in [20].…”