$H_{n+1}(K(G,n); \mathbb{Z}) = 0$ for $n > 1$

For \(n>1\), choose an \((n-1)\)-connected Moore space \(M(G,n)\), so \[ \widetilde H_i(M(G,n);\mathbb Z)= \begin{cases} G,&i=n,\\ 0,&i\ne n. \end{cases} \] By Hurewicz, \(\pi_n(M(G,n))\cong G\).

Construct a \(K(G,n)\) from \(M(G,n)\) by killing \(\pi_{n+1},\pi_{n+2},\dots\) successively. To kill \(\pi_k\) one attaches \((k+1)\)-cells. Thus the first cells added have dimension \(n+2\), and all later cells have still larger dimension. No \((n+1)\)-cells are added.

Initially \[ H_{n+1}(M(G,n))=0. \] Attaching cells of dimension \(n+2\) can only quotient \(H_{n+1}\) by images of new cellular boundaries; it cannot create a new \((n+1)\)-cycle. Cells of still higher dimension do not affect \(H_{n+1}\). Therefore the resulting Eilenberg–MacLane space satisfies \[ \boxed{H_{n+1}(K(G,n);\mathbb Z)=0\qquad(n>1).} \]