In Finite Groups of Lie Type written by Carter, a $BN$-pair of a group is defined to be two subgroups $B$ and $N$ such that

  1. $G$ is generated by $B$ and $N$.
  2. $B\cap N$ is normal in $N$.
  3. $N/(B\cap N)=W$ is generated by a set of elements $s_i$ with $s_i^2=1$.
  4. Let $n_i \in N$ map to $s_i \in W$. Then $n_i B n_i \neq B$.
  5. $n_iBn \subseteq Bn_iB \cup BnB$.

Let $G$ be any linear algebraic group, then $G$ is an algebraic group wit a split $BN$-pair if $G$ has closed subgroups $B,N$ satisfying:

  1. $B,N$ form a $BN$-pair in $G$.
  2. $B=U(B\cap N)$ is the semidirect product of a closed nomal unipotent group $U$ and a closed commutative subgroup $B \cap N$, all whose elemets are semisimple.
  3. $\cap_{n \in N}nBn^{-1}=B \cap N$.

Intuitively, $B$ and $N$ are a Borel subgroup and the normalizer of a torus in $G$. But in order to understand this definiton, I should have an idea when the conditons are not satisfed. Would anyone please give me some example of algebraic groups witout a split $BN$-pair? What is thc essnce of this definiton? Thanks very much.


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