On Units of Group Algebras of 2-Groups of Maximal Class

Abstract

Abstract. We determine the number of elements of order two in the group of normalized units V (F 2 G) of the group algebra F 2 G of a 2-group of maximal class over the field F 2 of two elements. As a consequence for the 2-groups G and H of maximal class we have that V (F 2 G) and V (F 2 H) are isomorphic if and only if G and H are isomorphic. Notation and results.Let G be a finite p-group and F p G its group algebra over the fieldis called the group of normalized units. Evidently V (F p G) is a p-group and its order is p |G|−1 .Let C = a | a 2 n = 1 be the cyclic group of order 2 n , where n ≥ 2. Consider the following extensions of C:which are the generalized quaternion group, the dihedral group and the semidihedral group respectively. It is well-known that a finite 2-group of maximal class coincides with one of these groups.

Isomorphism problem of Unitary Subgroups of Group Algebras

“…Berman [7] gave a positive answer for his question for finite abelian p-groups. Sandling [25] generalized the previous result, proving that if G is a finite abelian p-group, then a subgroup of V (F G), independent as a subset of the vector space F G, is isomorphic to a subgroup of G. For finite non-abelian p-groups (p > 2) with cyclic Frattini subgroup as well as for the class of maximal 2-groups the Berman's question has also a positive solution [4,5].…”

The structure of the unit group of some group algebras

“…The Johnson's question was affirmatively confirmed for nonabelian groups in the following cases: (i) the group of exponent p and order p 3 [15,Theorem 7]; (ii) G is a finite p-group (p is an odd prime) with Frattini subgroup of order p [4]; and (iii) G is the modular 2-group of order 16 and F is the field of 2 elements [14,Theorem 2]. The structure of elements of order two in V (FG), where G is a 2-group of maximal class and F is the field of elements two, was described in [5].…”

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.…”

“…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]).…”