# Relation between reflection group and coxeter group

Reflection group is defined see https://en.wikipedia.org/wiki/Reflection_group. An abstract Coxter group is defined to have generators $s_1$, $s_2$, ..., $s_n$ and relations $s^2_i=e$, $(s_is_j)^{m_{ij}}=e$ for some $2\leq m_{ij}\leq \infty$.

I don't know why viewed as an abstract group, every reflection group is a Coxeter group? Can somebody give me an example to explain this? Thanks in advance.

• It says on the Wikipedia page there that every reflection group is a specific example of a Coxeter group. – Nick Mar 21 '16 at 21:07
• @Nick Please you tell me why every reflection group is a specific example of a Coxeter group. Can you give me an example? Thanks in advance. – bing Mar 22 '16 at 1:34

## 1 Answer

We have $(s_i)^2 = e$ because if we repeat the same reflection twice in a row we end up back where we started.

Since $s_i s_j$ is an element of a group, it has an order (possibly infinity), which we denote by $m_{ij}$.

So in any reflection group, conditions of this form are at least satisfied. It remains to show that they're sufficient to completely define the group. As far as I know this is not so easy to explain, but Coxeter does this by characterizing the possible fundamental domains of a reflection group and then exploiting their polyhedral geometry. See:

where this is Theorem 8.