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Consider the abelian group $G$ generated by $a$, $b$ and $c$ and determined by the following relations

\begin{aligned} 3 a+9 b+9 c &=0 \\-3 b+9 c &=0 \end{aligned}

determine the isomorphism type of the group $G$.

I should use The Fundamental Theorem of Finite Abelian Groups. I think I should determine the orders of the generators, but i I do not see how.

Any help would be appreciated

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The relations give $b=3c$ and $a=-12c$. Therefore we can omit $a$ and $b$ from the presentation and we have no relations remaining for $c$:

$G=<c>$.

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  • $\begingroup$ How do you get $b=3c$? $\endgroup$ – lhf Mar 19 at 0:08
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    $\begingroup$ Using the second relation; we can assume $3 \neq 0$. $\endgroup$ – Lukas Kofler Mar 19 at 0:09
  • $\begingroup$ If I add 3 times the second relation to the first relation we get t $3(a+12c)=0$, can we conclude that $a=-12c$? I am missing something? $\endgroup$ – Usser123456798 Mar 19 at 2:18
  • $\begingroup$ That‘s one way to do it, that‘s correct! $\endgroup$ – Lukas Kofler Mar 19 at 3:06

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