Symplectic groups

The symplectic group Sp(2m,𝔽q) is the group of 2m × 2m matrices over 𝔽q preserving a non-degenerate alternating bilinear form (⟨v, v⟩ = 0, hence ⟨u, v⟩ = −⟨v, u⟩). Such a form exists only in even dimension, and all are equivalent, so the group is unique up to conjugacy.

• Order — |Sp(2m,𝔽q)| = qm² ∏i=1m (q2i − 1).
• Inside SL — every symplectic matrix has determinant 1: Sp(2m,𝔽q) ≤ SL(2m,𝔽q).
• Center — the scalars preserving the form are {±I} (λ² = 1), of order gcd(2, q − 1); the quotient is PSp.
• Generators — the symplectic transvections x ↦ x + λ⟨x, v⟩v.
• Action — transitive on the non-zero vectors of 𝔽q2m (Witt's theorem).

Small cases

• Dimension 2 — the alternating form is the determinant form, so Sp(2,𝔽q) = SL(2,𝔽q): Sp(2,𝔽2) ≅ S₃, Sp(2,𝔽3) = SL(2,𝔽3) (order 24), Sp(2,𝔽5) = SL(2,𝔽5) (order 120).
• Sp(4,𝔽2) ≅ S₆ — order 720; this exceptional isomorphism underlies the outer automorphism of S₆.

Part of the wider family of matrix groups: the alternating- form sibling of SL, with projective quotient PSp.

Wikipedia: Symplectic group