Topics in the isomorphism of group rings

Abstract

Introduction: Let R be a ring and let G be a group. The group ring R(G) of G over R is the free left R-module over the set of elements of G as a basis in which the multiplication induced by G is extended linearly to R(G) , [12]. A twisted group ring Ry (G) of G over R is an R-algebra with basis {ḡ |g ε G} and with an associative multiplication ḡ ħ = Y(g, h) ḡħ for all g, h ε G , where y(g, h) is a unit in the centre of R , [13]. In [5] Higman proved that the only units of finite order in the group ring R(G), where R is the ring of rational integers and G is a finite abelian group, are ± g, g ε G . In [16] Sehgal proved that the only units of finite order in the group ring R(G), where R is the ring of rational integers and G is an arbitrary abelian group, are ± t where t is a torsion element of G. Moreover in [16] he proved that the units of R(G), where R is an integral domain and G is a torsion-free abelian group, are of the form r g where r is a unit in R and g ε G. Also in [15] he proved that the units of R(<x>), where R is a commutative ring with no non-zero nilpotents and no non-trivial idempotents and <x> is an infinite cyclic group, are of the form r g where r is a unit in R and g ε <x>. In [17] Zariski and Samuel studied R-automorphisms of the polynomial rings R[x], (that is, automorphisms of R[x] which restrict to the identity mapping on R) where R is an integral domain. In [3] Gilmer determined R-automorphisms of the polynomial rings R[x] where R is a commutative ring. In [2] Coleman and Enochs studied the corresponding results in general. In [9] Parmenter studied R-automorphisms of the group ring R(<x>) where <x> is an infinite cyclic group and he determined necessary and sufficient conditions that x → Σ aixi induces an R-automorphism of R(<x>). He also studied the units of R(G) where R is a commutative ring and G is a right-ordered group. This thesis consists of five chapters. Chapter 1 contains some well known results and definitions that are needed in this thesis. In Chapter 2 we extend some ideas of [9] to a twisted group ring RY (<x>) where <x> is an infinite cyclic group and we determine a necessary and sufficient condition that x̄ → Σ l aixi induces an R-automorphism of RY (<x>) . Chapter 3 studies R-automorphism of R(G) where R is either a field or a ring with a unique proper ideal and G is a finitely generated torsion-free abelian group. In Chapter 4 we determine the units and study the K-automorphisms of K(<x> x <y>) where K is a field and <x> is an infinite 2 cyclic group, y2 = 1. In [10] Passman proved that the group algebras of all non-isomorphic p-groups of order at most p4 over the prime field of p elements are non-isomorphic. In Chapter 5 we attempt to find the corresponding results for the p-groups of order p 5, but the problem is still open.