Find the isomorphism type

Question

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

Answer 1

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 $c$:

$G=<c>$.