Borel subgroups

The Borel subgroup B(n,𝔽q) of GL(n,𝔽q) is the group of all invertible upper-triangular matrices — the diagonal torus extended by the unitriangular group: B(n,𝔽q) = T ⋉ UT(n,𝔽q) with T ≅ (Cq−1)ⁿ (a semidirect product, the torus scaling each entry above the diagonal).

• Order — |B(n,𝔽q)| = (q − 1)ⁿ · qn(n−1)/2.
• Solvable and self-normalizing in GL(n,𝔽q).
• Flag stabilizer — B(n,𝔽q) is the stabilizer of the standard complete flag ⟨e₁⟩ ⊂ ⟨e₁, e₂⟩ ⊂ ⋯; all complete-flag stabilizers are conjugate to it.
• Parabolic subgroups — the subgroups of GL(n,𝔽q) containing B(n,𝔽q) are exactly the 2ⁿ⁻¹ block-upper-triangular (parabolic) subgroups, one per composition of n; for n = 2 the only one besides B is GL(2,𝔽q) itself, so there B is maximal.
• Bruhat decomposition — GL(n,𝔽q) is the disjoint union of the n! double cosets B w B, one per permutation matrix w.
• Over 𝔽2 the torus is trivial: B(n,𝔽2) = UT(n,𝔽2).

Small cases

• B(2,𝔽3) — order 12, isomorphic to D₆: the torus C₂ × C₂ acts on UT(2,𝔽3) ≅ C₃ by inversion, with the scalars {±I} acting trivially.
• B(2,𝔽5) — order 80; B(3,𝔽3) — order 216.

The structure torus ⋉ unipotent is the model case of a Borel subgroup of an algebraic group.

Part of the wider family of matrix groups.