Large Groups of Unit of Integral Group Rings of Finite Nilpotent Groups

This paper surveys recent results regarding large subgroups of units in integral group rings of nilpotent groups, exibiting families of generators in several cases.nul

Torsion matrices over commutative integral group rings

Let ZA be the integral group ring of a finite abelian group A, and n a positive integer greater than 5. We provide conditions on n and A under which every torsion matrix U, with identity augmentation, in GLn(ZA) is conjugate in GLn(QA) to a diagonal matrix with group elements on the diagonal. When A is infinite, we show that under similar conditions, U has a group trace and is stably conjugate to such a diagonal matrix

Finite Matrix Groups over Nilpotent Group Rings

AbstractWe study groups of matricesSGLn(ZΓ) of augmentation one over the integral group ring ZΓ of a nilpotent group Γ. We relate the torsion ofSGLn(ZΓ) to the torsion of Γ. We prove that all abelianp-subgroups ofSGLn(ZΓ) can be stably diagonalized. Also, all finite subgroups ofSGLn(ZΓ) can be embedded into the diagonal Γn<SGLn(ZΓ). We apply matrix results to show that if Γ is nilpotent-by-(Π′-finite) then all finite Π-groups of normalized units in ZΓ can be embedded into Γ

A Conjecture of Brian Hartley and developments arising

Around 1980 Brian Hartley conjectured that if the unit group of a torsion group algebra FG satisfies a group identity, then FG satisfies a polynomial identity. In this short survey we shall review some results dealing with the solution of this conjecture and the extensive activity that ensued. Finally, we shall discuss special polynomial identities satisfied by FG (or by some of its subsets) and the corresponding group identities satisfied by its unit group (or by some of its subsets)

∗-group identities on units of group rings

Analogous to ∗-polynomial identities in rings, we introduce the concept of ∗- group identities in groups. When F is an infinite field of characteristic different from 2, we classify the torsion groups with involution G so that the unit group of FG satisfies a ∗-group identity. The history and motivations will be given for such an investigation

Topics in Group Rings

VI, 251 tr.; 23 cm