0
$\begingroup$

Let $G$ be the group generated by $a,b,c$ with relations \begin{align} a^2=1, b^2=1, c^2=1, ac=ca, bc=cb, abab=baba. \end{align} What is this group? I think that the group has order $16$. Is it correct?

$\endgroup$
  • $\begingroup$ Why do you think that the order is $16$? $\endgroup$ – Arnaud Mortier Jul 8 '18 at 12:20
3
$\begingroup$

Clearly $c$ is central, so $G=\left< a,b\right>\times\left<c\right>$. But the relations between $a$ and $b$ are $a^2=b^2=(ab)^4=1$ so that $\left<a,b\right>$ is dihedral of order $8$.

$\endgroup$

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.