On the finiteness of the subgroup of homotopy self-equivalences inducing the identity on homology

For a rationally elliptic space [Formula: see text], we denote its group of self-homotopy equivalences as [Formula: see text], and its subgroup consisting of elements inducing the identity on the homology groups as [Formula: see text]. This paper aims to investigate how the homotopy groups of [Formula: see text] influence the finiteness of [Formula: see text].