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

Let $G$ be a finite group and $\mathrm U(Z(\mathbf ZG))$ be the group of units of the center $Z(\mathbf ZG)$ of the integral group ring $\mathbf ZG$ (the central unit group of the ring $\mathbf ZG$). The purpose of the present work is to study the ranks $r_n$ of groups $\mathrm U(Z(\mathbf Z\mathrm A_n)$, i.e., of central unit groups of integral group rings of alternating groups $\mathrm A_n$. We shall find all values $n$ for $r_n=1$ and propose an approach how to describe the groups $\mathrm U(Z(\mathbf Z\mathrm A_n))$ in these cases, and we will present some results of calculations of $r_n$ for $n\leq600$.