Unit Group

We discuss the structure of the unitary subgroup V * (F 2 q D 2 n ) of the group algebra F 2 q D 2 n , where D 2 n = x, y | x 2 n−1 = y 2 = 1, xy = yx 2 n−1 −1 is the dihedral group of order 2 n and F 2 q is any finite field of characteristic 2, with 2 q elements. We will prove that V * (F 2 q D 2 n ) ∼ = C (3.(2 n−2 )−1)q 2 C q 2 , see Theorem 3.1.

For a finite abelian group A, let us denote the unit group of its integral group ring by U (ZA). The rank of torsion free part of U (ZA) is determined by Ayoub and Ayoub as ρ = 1 2 (|A| + 1 + n 2 − 2l) where n 2 is the number of elements of A of order 2 and l is the number of cyclic subgroups of of A. Here ρ mainly depends on the number of cyclic subgroup of A. In this study, we have first computed the rank of U (ZC n ) by using prime factorization of |C n | . Then by extending cyclic subgroup C n by B = C 2 , C 3 , C 4 or K 4 , we have computed the rank of U (ZA) where A = C n × B. Finally, we have given rank tables for both cyclic abelian and non-cyclic abelian groups of small orders.