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

We prove that the ranks of central unit groups of integral group rings of alternating groups of degrees greater than 38 are at least 11. The presented tables contain ranks of all central unit groups of integral group rings of alternating groups of degrees at most 200. In particular, for every $r\in\{0,\dots,10\}$, we obtain the complete list of integers $n$ such that the central unit group of the integral group ring of the alternating group of degree $n$ has rank $r$.