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.
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.
