A study on units in group rings

Abstract

The primary objective of this thesis is to investigate the unit group of group rings and address the normal complement problem in the unit group. Firstly, we assume that F is finite field of characteristic 2 and investigate the existence of normal complements for the dihedral group D4m of order 4m and the alternating group A4, where m is an odd integer greater than or equal to 3. A normal complement for S4 in V(FS4) over a field F containing exactly two elements has been found. Further, let Zn be the ring of integers modulo n. We use Ct, Em, and Fr,s to respectively denote the cyclic group of order t, the elementary abelian 2-group of order 2m, and an abelian group of exponent 4 with order 2r4s. We find the generators of the normalized unit group V(ZnC2) and solve the normal complement problem in V(ZnC2). We also provide a normal complement of Em in V(Z2nEm). Furthermore, we determine the structure of V(ZpnFr,s) for an odd prime p and establish that Fr,s does not have a normal complement in V(ZpnFr,s). Moreover, we give the structure and generators of the unit group U(ZnC3). Lastly, we provide the structure of U(ZnTm), where Tm is the elementary abelian 3-group of order 3m and gcd(n,3) = 1.