Showing a quotient group $G/\langle x_0\rangle$ to be a direct sum of cyclic groups There is an infinite p-primary group which is not isomorphic to $\mathbf{Z}_{p^∞}$ :
$G=\langle x_0,x_1,\ldots\mid px_0, x_0-p^nx_n\rangle$ for all $n\geq 1$. 
Can I ask any hints, for showing that $G/\langle x_0\rangle$ is a direct sum of cyclic groups? 

Can you describe $G/\langle x_0\rangle$ in a similar fashion as your defined $G$? Or if you want : Knowing that in $G$ you have $p x_0 = 0$, $p^n x_n = x_0$ and that $x_0 = 0$ in $G/\langle x_0\rangle$, you should get simpler relations. 
