BN-pair

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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_iB and v_iBv_i \not \subseteq B

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