Let A be a finite abelian group and G = A x 〈b〉, b2 = 1, ab = a-1, ∀a ∈ A. We find generators up to finite index of the unitary subgroup of ZG. In fact, the generators are the bicyclic units. For an arbitrary group G, let B2(ZG) denote the group generated by the bicyclic units. We classify groups G such that B2(ZG) is unitary.
We show that any semiartinian -regular ring is unit-regular; if, in addition, is subdirectly irreducible then it admits a representation within some inner product space.
In [3], the authors initiated a technique of using affine representations to study the groups of units of integral group rings of crystallographic groups. In this paper, we use this approach for some special classes of crystallographic groups. For a first class of groups we obtain a normal complement for the group inside the group of normalized units. For a second class of groups we show that the Zassenhaus conjectures ZC1 and ZC3 are valid. This generalizes the results known for the infinite dihedral...
2000 Mathematics Subject Classification: 20C05, 16U60, 16S84, 15A33.The Structure of the Unit Group of the Group Algebra of the group D10 over any field of characteristic 5 is established in terms of split extensions of cyclic groups.