Units Of Integral Group Rings Research Articles

Integral group rings of frobenius groups and the conjectures of H.J. Zassenhaus

The conjecture of H.J. Zassenhaus for finite subgroups of units of integral group rings. restricted to p-subgroups, is proved for finite Frobenius groups when p is an odd prime. The result for 2-subgroups is established for those Frobenius groups that cannot be mapped homo-morphically onto S$sub:5$esub:. The conjecture in its full strength is proved for A5, S5 and SL(2.5).

Torsion units in integral group rings, conjugacy classes globally versus locally

Torsion units in integral group rings, conjugacy classes globally versus locally

Central units of integral group rings of nilpotent groups

In this paper a finite set of generators is given for a subgroup of finite index in the group of central units of the integral group ring of a finitely generated nilpotent group.

On the Torsion Units in Integral Group Rings of Dihedral 2-Groups

On the Torsion Units in Integral Group Rings of Dihedral 2-Groups

Generators of Large Subgroups of Units of Integral Group Rings of Nilpotent Groups

Let G be a finite nilpotent group so that all simple components ( D) n × n , n ≥ 2 of Q G satisfy the congruence subgroup theorem. Suppose that for all odd primes p dividing | G| the Hamiltonian quaternions H split over the pth cyclotomic field Q (ζ p ). Then new units B 3 are introduced so that 〈 B 1, B 2, B ′ 2, B 3〉 is of finite index in U( Z G).

Units in Integral Group Rings for Order pq

AbstractFor any finite abelian group A, let Ω(A) denote the group of units in the integral group ring which are mapped to cyclotomic units by every character of A. It always contains a subgroup Y(A), of finite index, for which a basis can be systematically exhibited. For A of order pq, where p and q are odd primes, we derive estimates for the index [Ω(A) : Y(A)]. In particular, we obtain conditions for its triviality.

Conjugates of units in integral group rings

Conjugates of units in integral group rings

Cyclotomic units over finite fields

Let ζ be a primitivesp-th root of unity for a primep>2, and consider the group Ω(ζ) of cyclotomic units in the ringR(ζ)=ℒ[ζ+ζ-1]. This paper deals with the image of Ω(ζ) in the unit group ofR(ζ)/qR(ζ), whereq is a prime ≠p. In particular, it obtains criteria for this image to be essentially everything, and a lower bound on the density of primesp (withq fixed) for which it cannot be. These results have a direct bearing on previous work about units in integral group rings for cyclic groups of orderpq.

Units of Integral Group Rings of Some Metacyclic Groups

AbstractIn this paper, we consider all metacyclic groups of the type 〈a,b | an - 1, b2 = 1, ba = aib〉 and give a concrete description of their rational group algebras. As a consequence we obtain, in a natural way, units which generate a subgroup of finite index in the full unit group, for almost all such groups.

Torsion units in integral group rings of solvable groups

A weaker version of the Zassenhaus conjecture for torsion units in integral group rings ZG is proved if G is either abelian-by-polycyclic or metabelian. As a consequence we obtain Bovdi's conjecture for torsion units in ZG for metabelian groups

Torsion units in integral group rings

Several special cases of the conjectures of Bovdi and Zassenhaus are proved. We also deal with special cases of the following conjecture: let α be a torsion unit of the integral group ring ZZG and m the smallest positive integer such that αm ∈G then, m is a divisor of the exponent of the quotient group G/Z(G) provided this exponent is finite

Torsion units in integral group rings

Let G = ⟨ a ⟩ ⋊ X G = \left \langle a \right \rangle \rtimes X where ⟨ a ⟩ \left \langle a \right \rangle is a cyclic group of order n , X n,X is an abelian group of order m m , and ( n , m ) = 1 (n,m) = 1 . We prove that if Z G \mathbb {Z}G is the integral group ring of G G and H H is a finite group of units of augmentation one of Z G \mathbb {Z}G , then there exists a rational unit γ \gamma such that H γ ⊆ G {H^\gamma } \subseteq G .

Jordan decomposition and hypercentral units in integral group rings

(1993). Jordan decomposition and hypercentral units in integral group rings. Communications in Algebra: Vol. 21, No. 1, pp. 25-35.

Torsion units in integral group rings of nilpotent metabelian groups

Torsion units in integral group rings of nilpotent metabelian groups

Construction of Units in Integral Group Rings of Finite Nilpotent Groups

Construction of Units in Integral Group Rings of Finite Nilpotent Groups

Torsion units in integral group rings.

Torsion units in integral group rings.

Construction of units in integral group rings of finite nilpotent groups

Let U be the group of units of the integral group ring of a finite group G. We give a set of generators of a subgroup B of U. This subgroup is of finite index in U if G is an odd nilpotent group. We also give an example of a 2-group such that B is of infinite index in U.

Describing units of integral group rings of some 2-groups

Describing units of integral group rings of some 2-groups

Construction of units in integral group rings of finite nilpotent groups

Let U be the group of units of the integral group ring of a finite group G. We give a set of generators of a subgroup B of U. This subgroup is of finite index in U if G is an odd nilpotent group. We also give an example of a 2-group such that B is of infinite index in U.

A theorem on units of integral group rings

Let G be a finite group and U = U(ℤG) be the unit group of the integral group ring ℤG. Let H be the normal subgroup of U generated by the unit groups of the cyclic subgroup rings. It is proved that, under certain restrictions on G, H has finite index in U. The main tools are a theorem of Bass concerning K1(ℤG) and a theorem of Margulis of discrete subgroups of Lie groups.