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 $c$:


  • $\begingroup$ How do you get $b=3c$? $\endgroup$ – lhf Mar 19 at 0:08
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.