$G = \langle a, b : a^3 = b^2 = (ab)^3 = e \rangle$. Find the order of G.

Question

$G$ = $\langle$$a, b : a^3 = b^2 = (ab)^3 = e \rangle$. What is |$G$|.

I don't even know where to begin for this question. Any help would be appreciated.

Answer 1

Probably the best way to go here is to bound the number of combinations of $a$ and $b$ as per Dave suggestion. Here is how you should begin, first by definition, any element of $g$ will be written as:

$$a^{n_1}b^{m_1}\dots a^{n_r}b^{m_r} $$ where $n_i$ and $m_i$ are in $\mathbb{Z}$. Now I claim that you can choose $n_i=0,1$ or $2$ and $m_i=0$ or $1$. Do you see why?

Then you need to justify that $ba^2$ can be replaced by $(ab)^2$ and then you should be able to conclude that $G$ is finite (you should even be able to make an explicit list of elements). The hardest part IMHO is to show that the remaining elements are not trivial and different from each other. The best way to do this is probably to find elements in a known group verifying these relations (Gerry Myerson gave a suggestion).