Showing a quotient group $G/\langle x_0\rangle$ to be a direct sum of cyclic groups

Question

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?

simply rewrite the presentation of $G$ by replacing $x_0$ with $0$ - you should see it then — user8268, May 07 '11 at 14:21

score 4 · Accepted Answer · edited May 07 '11 at 15:37

4

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.

edited May 07 '11 at 15:37

Asaf Karagila

393,674

answered May 07 '11 at 14:24

Joel Cohen

9,289

I am really thinking of showing that this quotient group is finite, so the problem will be easy by applying "Basis Theorem" next. – Nancy Rutkowskie May 07 '11 at 14:40
I don't think this is necessary. You can get a very explicit decomposition if you write the presentation of $G/<x_0>$. – Joel Cohen May 07 '11 at 14:48
For future reference $<,>$ are interpreted as relations, you want \langle and \rangle instead. – Asaf Karagila May 07 '11 at 15:38

Showing a quotient group $G/\langle x_0\rangle$ to be a direct sum of cyclic groups

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