BN-pair

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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Definition

Definition with symbols

Let $G$ be a group. A pair of subgroups $B, N$ of $G$ is termed a BN-pair if it satisfies the following conditions:

$G$ is generated by two subgroups $B$ and $N$
$H : = B \cap N ◃ N$ , viz the intersection is normal in the second subgroup
$W = N / H$ is generated by involutions $w_{1}, w_{2}, \dots, w_{m}$
If $v_{i}$ is a coset representative of $w_{i}$ , then for each $v \in N$ and every $i$ :

$v B v_{i} \subseteq B v B \cup B v v_{i} B$ and $v_{i} B v_{i} \subseteq̸ B$

Such a setup is also called a Tits system of rank $m$ .