For a homotopy associative \(H\)-space \(X\) with homotopy inverse, let \(\varphi_{X,1}=\operatorname{id}_X\) and let \(\varphi_{X,2}:X^2\to X\) denote the commutator map. For \(n\geq 3\), let \(\varphi_{X,n}:X^n\to X\) be the \(n\)-th commutator map defined by \(\varphi_{X,n}=\varphi_{X,2}\circ (\operatorname{id}_X\times \varphi_{X,n-1})\). An \(H\)-space \(X\) is called homotopy nilpotent of class \(n\) if \(\varphi_{X,n+1}\) is null homotopic (i.e. \(\varphi_{X,n+1}\simeq *\)) but \(\varphi_{X,n}\) is not. In this case we write \(\operatorname{nil}X=n\). An \(H\)-space \(X\) is called homotopy nilpotent if \(\operatorname{nil}X<\infty\).
In this paper, the author investigates whether the loop space \(\Omega M(A,n)\) is homotopy nilpotent, where \(A\) is an abelian group and \(M(A,n)\) denotes the Moore space of type \((A,n)\). For \(n\geq 2\) he proves that \(\Omega M(A,n)\) is homotopy nilpotent if and only if \(A\) is a subgroup of \(\mathbb{Q}\). Moreover, he also studies the homotopy nilpotency of the loop space \(\Omega M(A,1)\) and computes \(\operatorname{nil}\Omega M(A,1)\) explicitly for several such examples.