Coxeter group

Remarks

The relation for $i = j$ says $s_i^2 = 1$. For $m_{ij} = 2$ the relation says that $s_i$ and $s_j$ commute. In $W(M)$ each $s_i$ has order exactly $2$, and $s_i s_j$ has order exactly $m_{ij}$. The matrix is often drawn as a Coxeter diagram: a vertex for each $i$, and an edge labelled $m_{ij}$ when $m_{ij} \ge 3$.

Example: the symmetric group $S_n$ is a Coxeter group. Take $s_i = (i \ i{+}1)$ for $1 \le i \le n - 1$, with $m_{i, i+1} = 3$ and $m_{ij} = 2$ for $\abs{i - j} \ge 2$. The relations hold: $(i \ i{+}1)(i{+}1 \ i{+}2)$ is a $3$-cycle, and disjoint transpositions commute. A theorem of Moore says that these relations define $S_n$.

Non-example: the cyclic group $\Z/4$ is not a Coxeter group. A Coxeter group is generated by elements of order $2$, so in its abelianization every element has order at most $2$. But $\Z/4$ is abelian, so it equals its abelianization, and it has an element of order $4$.