5
$\begingroup$

I would like help with the following problem:

Let $G$ be the subgroup of $GL_3(\mathbb{C})$ generated by the three matrices $$ A=\begin{pmatrix} 0&0&1\\0&1&0\\1&0&0\end{pmatrix}\;,\quad B=\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}\;,\quad C=\begin{pmatrix} i&0&0\\0&1&0\\0&0&1\end{pmatrix}$$ 1. Compute the order of $G$. 2. Find a matrix $G$ of largest possible order (as an element of $G$) and compute this order. 3. Compute the number of elements in $G$ with this largest order.

So I've found the relations $A^2=B^3=C^4=I$, so I know my group so far consists of $\{I,A,B,B^2,C,C^2,C^3\}$, but I don't know how to proceed from here. I know the group is not abelian, but I just thought I'd try listing all two element products, so I have 30=6(5) choices for that (assuming, which I haven't checked, they are all distinct). I am just wondering if there is a more concrete way to approach this problem. I'd also like help with the other parts afterward. I can see that $BC$ and $CB$ have order 12, but I'm not sure what to do next.

  • 1
    Have you tried the Todd-Coxeter algorithm? Alternately, there is a not-terribly-complicated finite group which contains $G$ as a subgroup (permutation matrices with entries in $\{ 1, -1, i, -i \}$) and you could try to compute the index of $G$ in this group.2012-08-20
  • 0
    I don't know for sure that this will work, but you might try computing commutators like $A^{-1}B^{-1}AB$. If you can express them all in the form $A^iB^jC^k$ then you can get a standard form for elements of the group, and that will give you the order. Even if it doesn't quite work out that way you may get enough relations to get the kind of information you want.2012-08-20
  • 0
    I'm not familiar with that algorithm, but I can try learning it and maybe it will lead to a solution. Would you mind please elaborating more on the other method with the permutation matrices?2012-08-21
  • 0
    @ Qiaochu Yuan, is this also obvious so you can shed light on it: https://math.stackexchange.com/questions/2731225/2018-04-10

2 Answers 2