let $G$ be an abelian torsion group. let $z\in H_i(K(G,1))$. throughout the coefficient group is $\mathbb Q$.
1) Why there exists a FINITE subcomplex $X$, such that $j:X\hookrightarrow K(G,1),\, j_*(x)=z$ for some $x\in H_i(X)$ ?
2) Why the image of $\pi_1(X)$ in $\pi_1(K(G,1))=G$ is finite?
MY guess:
1) $G=\oplus_p G_p$ where $G_p$ ranges over all $p$-subgroups of $G$ so by Kunneth $H_i(K(G,1))=\oplus_{i_1+\cdots+i_l=i}H_{i_1}(K(G_{p_1},1) \otimes \cdots\otimes H_{i_l}(K(G_{p_l},1 ) $ and I don't know what does it mean for $z$ to be in this decomposition!!
2) if $X$ is a finite CW that implies that its $\pi_1$ is finitely generated and not finite !!! and what finite CW means here anyway: having a finite number of cells at each dimension or having no cells beyond a dimension $n$ so that $X$ equals its $n$-skeleton?