Unit Group

The problem of describing the group of units U(ZG) of the integral group ring ZG of a finite group G has attracted a lot of attention and providing presentations for such groups is a fundamental problem. Within the context of orders, a central problem is to describe a presentation of the unit group of an order O in the simple epimorphic images A of the rational group algebra QG. Making use of the presentation part of Poincaré's polyhedron theorem, Pita, del Río and Ruiz proposed such a method for a large family of finite groups G and consequently Jespers, Pita, del Río, Ruiz and Zalesskii described the structure of U(ZG) for a large family of finite groups G. In order to handle many more groups, one would like to extend Poincaré's method to discontinuous subgroups of the group of isometries of a direct product of hyperbolic spaces. If the algebra A has degree 2 then via the Galois embeddings of the centre of the algebra A one considers the group of reduced norm one elements of the order O as such a group and thus one would obtain a solution to the mentioned problem. This would provide presentations of the unit group of orders in the simple components of degree 2 of QG and in particular describe the unit group of ZG for every group G with irreducible character degrees less than or equal to 2. The aim of this paper is to initiate this approach by executing this method on the Hilbert modular group, i.e., the projective linear group of degree two over the ring of integers in a real quadratic extension of the rationals. This group acts discontinuously on a direct product of two hyperbolic spaces of dimension two. The fundamental domain constructed is an analogue of the Ford domain of a Fuchsian or a Kleinian group.

In this article we construct free groups and subgroups of finite index in the unit group of the integral group ring of a finite non-abelian group G for which every non-linear irreducible complex representation is of degree 2 and with commutator subgroup G a central elementary abelian 2-group. * Research partially supported by the Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium) and Bilateral Scientific and Technological Cooperation BWS 05/07 (Flanders-POland). † Postdoctoraal Onderzoeker van het Fonds voor Wetenschappelijk Onderzoek-Vlaanderen. ‡ Research partially supported by the Fundación Séneca of Murcia and D.G.I. of Spain.

We provide a lower bound for the efficiency of polarization or coherence transfer between quantized states under unitary transformations. Mathematically the problem is the determination of the C-numerical radius of A for certain nilpotent matrices C and A. The presented lower bound is conjectured to be exact as it coincides with numerical data provided in [U. Helmke et al.,

By a finite abelian group G, one can construct the integral group ring ZG. Let us denote the unit group of ZG by U (ZG). The rank of torsion-free part of unit group is determined by Ayoub and Ayoub [4] as ρ = 1 2 (|G| + n 2 + 1 − 2l) with respect to the result of Higman [5]. Here n 2 denotes the number of elements of order 2 in G and l denotes the number of all distinct cyclic subgroups of G. Kelebek and Bilgin have given the relations on the rank of unit group U (ZA) for the group A = C n × B where B = C 2 , C 3 , C 4 and K 4 [1].In this paper, we get the rank of torsion-free part of unit group for C * n = C n × B where B = C 6 by constructing order table of elements in C * n according to some n = p k , p k q.

In this note, we have given the center Z(V * (F 2 m M 16 )) of unitary units subgroup V * (F 2 m M 16 ) of group algebra F 2 m M 16 , where M 16 = x, y | x 8 = y 2 = 1, xy = yx 5 is the Modular group of order 16 and F 2 m is any finite field of characteristic 2, with 2 m elements. The structure of the unitary unit subgroup V * (F 2 m M 16 ) of the group algebra F 2 m M 16 , is also described, see Theorem 3.1.

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.