The ranks of central unit groups of integral group rings of alternating groups

Abstract

Let G be a finite group and U(Z(Z G)) be the group of units of the center Z(Z G) of the integral group ring Z G (the central unit group of the ring Z G). The purpose of the present work is to study the ranks r n of groups U(Z(ZAn)), i.e., of central unit groups of integral group rings of alternating groups A n . We shall find all values n for r n = 1 and propose an approach on how to describe the groups U(Z(ZAn)) in these cases, and we will present some results of calculations of r n for n ≤ 600.