# Find the isomorphism type

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

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 $$c$$:

$$G=$$.

• How do you get $b=3c$? – lhf Mar 19 at 0:08
• Using the second relation; we can assume $3 \neq 0$. – Lukas Kofler Mar 19 at 0:09
• 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? – Usser123456798 Mar 19 at 2:18
• That‘s one way to do it, that‘s correct! – Lukas Kofler Mar 19 at 3:06