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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?

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    simply rewrite the presentation of $G$ by replacing $x_0$ with $0$ - you should see it then2011-05-07

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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.

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    For future reference <,> are interpreted as relations, you want \langle and \rangle instead.2011-05-07