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

$vBv_{i}\subseteq BvB\cup Bvv_{i}B$ and $v_{i}Bv_{i}\not \subseteq B$

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