Just took my final exam and I wanted to see if I answered this correctly:
If $A$ is a Abelian group generated by $\left\{x,y,z\right\}$ and $\left\{x,y,z\right\}$ have the following relations:
$7x +5y +2z=0; \;\;\;\; 3x +3y =0; \;\;\;\; 13x +11y +2z=0$
does it follow that $A \cong Z_{3} \times Z_{3} \times Z_{6}$ ?
I know if we set $x=(1,0,2)$, $y=(0,1,0)$ and $z=(2,1,5)$ then this is consistent with the relations and with $A \cong Z_{3} \times Z_{3} \times Z_{6}$