Units in finite dihedral and quaternion group algebras

Abstract

Let F q G be the group algebra of a finite group G over F q ¼ GFðqÞ. Using the Wedderburn decomposition of F 2 k D 2n =JðF 2 k D 2n Þ, we establish the structure of the unit group of F 2 k G when G is either D 4n , the dihedral group of order 4n or Q 4n , the generalized quaternion group of order 4n; n odd.

Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$

“…The dihedral group D 12 = x, y | x 6 = y 2 = 1, yx = x 5 y . The structure of the unit group of the finite group algebra F D 12 has been studied in [9,16,25] whereas the structure of unit group of F (C 2 × D 12 ) is described in [12,18,23]. In this section, we establish the structure of the unit group of F (C n × D 12 ).…”

“…Let F G be the group algebra of a finite group G over a finite field F of characteristic p having q = p k elements. Let U (F G) be the unit group of F G and let J(F G) be the Jacobson radical of F G. If V = 1 + J(F G), then U (F G) ∼ = V U (F G/J(F G)) [16]. A good description of the structure of U (F G) has applications in various areas like the group ring cryptography [10] and the combinatorial number theory [5], etc.…”

“…The groups D 2n , n ≥ 3 and Q 4n , n ≥ 2 also contain a normal cyclic subgroup of index 2 and the unit groups of the group algebras of these groups and their extensions have been extensively studied [6,7,9,11,13,14,16,17,18,21,23,25,26]. In this paper, we aim to contribute in this direction further by describing the structures…”

“…The structures of the unit groups of F Q 12 and F (C 2 × Q 12 ) have been studied in [9,18,25,26]. Also, the unit groups of F D 12 and F (C 2 × D 12 ) have been obtained in [9,16,18,23,25], whereas U (F A 4 ) is given in [8,24]. Throughout the paper, C k n is the direct product of k copies of C n , F n is the extension field of F of degree n and GL(n, F ) is the general linear group of degree n over F .…”