nLab BN-pair

Definition

A pair $B,N$ of subgroups of a group $G$ is a BN-pair if the following holds:

• the union $B\cup N$ generates $G$,

• the intersection $B\cap N$ is a normal subgroup of $N$,

• the quotient group $W \coloneqq N/(B\cap N)$ is generated by a set $S$ of involutions,

• for each $s\in S,w\in W$ we have $B s B \cdot B w B\subseteq B s w B\cup B w B$

• for $s\in S$, $B s B\cdot B s B \neq B$

Note that the set $S$ is uniquely determined by these axioms, and that $(W,S)$ is a Coxeter system.

References

• Anders Bjorner, Francesco Brenti, Combinatorics of coxeter groups, Springer 2005 (doi:10.1007%2F3-540-27596-7)