Matrix groups

A matrix group is a group of invertible n × n matrices over a finite field 𝔽q (or a finite ring). Fixing a determinant, a bilinear / Hermitian / quadratic form, a triangular shape, or adding a translation part carves out the families below. Each operator is its own named family; the parameters (n, q) are part of the name (GL(2,𝔽3), PSL(2,𝔽7)).

One size formula underlies all of them:

|GL(n,𝔽q)| = (qⁿ − 1)(qⁿ − q) ⋯ (qⁿ − qⁿ⁻¹) = qn(n−1)/2 ∏i=1n (qⁱ − 1).

The scalar matrices λI form the center of GL(n,𝔽q), isomorphic to 𝔽q× ≅ Cq−1; quotienting by scalars produces the projective groups.

Linear groups

GL — general linear

All invertible n × n matrices over 𝔽q — equivalently Aut(𝔽qn).

• Order qn(n−1)/2 ∏i=1n (qⁱ − 1); e.g. |GL(2,𝔽2)| = 6, |GL(2,𝔽3)| = 48, |GL(3,𝔽2)| = 168.
• Examples GL(2,𝔽2) ≅ S₃, permuting the three nonzero vectors of 𝔽2²; GL(3,𝔽2) ≅ PSL(2,𝔽₇), simple of order 168.
• Properties the determinant GL(n,𝔽q) → 𝔽q× is onto with kernel SL, so GL/SL ≅ Cq−1; the center is the scalars ≅ Cq−1.

SL — special linear

The determinant-1 subgroup of GL(n,𝔽q).

• Order |GL(n,𝔽q)| / (q − 1); e.g. |SL(2,𝔽3)| = 24, |SL(2,𝔽5)| = 120.
• Examples over 𝔽2 every determinant is 1, so SL(2,𝔽2) = GL(2,𝔽2) ≅ S₃; SL(2,𝔽3) is the binary tetrahedral group Q₈ ⋊ C₃ = 2·A₄; SL(2,𝔽5) is the binary icosahedral group 2·A₅, the smallest perfect group that is not simple.
• Properties center = scalars λI with λⁿ = 1, of order gcd(n, q − 1); quotient by it gives PSL.

PGL — projective general linear

GL(n,𝔽q) modulo its scalar center — the faithful action on the (qⁿ − 1)/(q − 1) points of projective space.

• Order |GL(n,𝔽q)| / (q − 1).
• Examples PGL(2,𝔽q) acts sharply 3-transitively on the q + 1 points of the projective line: PGL(2,𝔽3) ≅ S₄, PGL(2,𝔽5) ≅ S₅.
• Properties contains PSL as a normal subgroup with PGL / PSL ≅ Cgcd(n, q−1), so PGL = PSL exactly when gcd(n, q−1) = 1.

PSL — projective special linear

SL(n,𝔽q) modulo its scalar center. The most important family: PSL(n,𝔽q) is simple for every n ≥ 2 and prime power q, except PSL(2,𝔽2) ≅ S₃ and PSL(2,𝔽3) ≅ A₄.

• Order |SL(n,𝔽q)| / gcd(n, q − 1).
• Examples PSL(2,𝔽4) ≅ PSL(2,𝔽5) ≅ A₅ (order 60, the smallest non-abelian simple group); PSL(2,𝔽7) ≅ PSL(3,𝔽2) (order 168, the second smallest); PSL(2,𝔽9) ≅ A₆ (order 360).
• Connection these form one infinite family of finite simple groups; the coincidences among small PSLs and alternating groups are the classic "exceptional isomorphisms".

Classical form groups

Sp — symplectic

2n × 2n matrices over 𝔽q preserving a non-degenerate alternating bilinear form.

• Order qn² ∏i=1n (q2i − 1).
• Examples in dimension 2 the alternating form is the determinant, so Sp(2,𝔽q) = SL(2,𝔽q); Sp(4,𝔽2) ≅ S₆ (order 720).
• Properties the center is {±I} of order gcd(2, q − 1); the quotient is PSp.

PSp — projective symplectic

Sp(2n,𝔽q) / {±I}, simple apart from small exceptions.

• Examples PSp(2,𝔽q) = PSL(2,𝔽q) (inheriting the PSL(2,𝔽2), PSL(2,𝔽3) exceptions); PSp(4,𝔽2) ≅ S₆; PSp(4,𝔽3) has order 25 920.
• Connection a second infinite family of finite simple groups, the classical groups of type Cₙ.

GU — general unitary

n × n matrices over 𝔽q² preserving a non-degenerate Hermitian form (relative to the order-2 field automorphism x ↦ xq).

• Order qn(n−1)/2 ∏i=1n (qⁱ − (−1)ⁱ); e.g. |GU(3,𝔽3)| = 24 192.
• Properties the determinant lands in the norm-1 subgroup of 𝔽q²× (order q + 1); its kernel is SU, so GU / SU ≅ Cq+1.

SU — special unitary

The determinant-1 subgroup of GU(n,𝔽q).

• Order |GU(n,𝔽q)| / (q + 1); e.g. |SU(3,𝔽3)| = 6048.
• Properties center = scalars of order gcd(n, q + 1); quotient gives PSU.

PSU — projective special unitary

SU(n,𝔽q) / Z, simple apart from small exceptions (PSU(2,𝔽q) ≅ PSL(2,𝔽q), PSU(3,𝔽2) solvable).

• Examples PSU(3,𝔽3) has order 6048; PSU(4,𝔽2) ≅ PSp(4,𝔽3) (order 25 920).
• Connection the classical simple groups of type ²Aₙ (a "twisted" family).

Orthogonal groups

The type is part of the name. Quadratic forms over a finite field come in two kinds in even dimension, so an even-dimensional member writes its Witt type immediately after the operator — GO+(4,𝔽3), Ω−(6,𝔽2) — while an odd-dimensional one writes none, the form there being unique up to scaling: SO(3,𝔽5). Plus type has maximal Witt index (an orthogonal sum of hyperbolic planes), minus type one less.

GO, SO, Ω and PΩ are four different groups. The ATLAS writes Onε(q) for the simple group PΩε(n, q), which is neither of the first two; names here say which group they mean.

GO — general orthogonal

Matrices over 𝔽q preserving a non-degenerate quadratic form. In odd dimension the form is unique (GO(2m+1,𝔽q)); in even dimension there are two inequivalent types, GO+(2m,𝔽q) and GO−(2m,𝔽q), distinguished by whether the form has maximal (+) or non-maximal (−) Witt index.

• Order 2 qm(m−1)(qm − ε)∏i<m(q2i − 1) in dimension 2m; minus type is the larger group.
• Properties contains the determinant-1 subgroup SO; over an even-order field SO = GO since every isometry then has determinant 1.
• Degenerate cases dimension 2 gives dihedral groups (GOε(2,q) ≅ Dq−ε), and odd dimension in even characteristic gives Sp(2m,𝔽q) — neither earns a name of its own.

SO — special orthogonal

The determinant-1 subgroup of GO, SO(3,𝔽5) / SO+(4,𝔽3) / SO−(4,𝔽3).

• Properties still has a further index-2 subgroup, the spinor kernel Ω; Ω is where the simple groups live.
• Example SO(3,𝔽q) ≅ PGL(2,𝔽q) for odd q.

Ω — omega

The kernel of the spinor norm inside SO — of the Dickson invariant in even characteristic. Written Ω(2m+1,𝔽q), Ω+(2m,𝔽q), Ω−(2m,𝔽q).

• Properties its projective quotient PΩ carries the simple orthogonal groups (types Bₙ, Dₙ, ²Dₙ); Ω itself may have a center of order 2.
• Connection in dimension at most 6 every orthogonal group is another classical group: Ω(3,𝔽q) ≅ PSL(2,𝔽q), Ω(5,𝔽q) ≅ PSp(4,𝔽q), Ω−(4,𝔽q) ≅ PSL(2,𝔽q²), Ω+(6,𝔽2) ≅ A₈.

Affine groups

AGL — affine general linear

Invertible affine maps x ↦ Ax + b of 𝔽qn — the translation group 𝔽qn extended by GL(n,𝔽q): AGL(n,𝔽q) = 𝔽qn ⋊ GL(n,𝔽q).

• Order qⁿ · |GL(n,𝔽q)|.
• Examples AGL(1,𝔽q) = 𝔽q ⋊ 𝔽q× (order q(q−1)) is sharply 2-transitive: AGL(1,𝔽3) ≅ S₃, AGL(1,𝔽4) ≅ A₄; AGL(2,𝔽2) ≅ S₄.
• Connection the point stabilizers and building blocks of the affine-type 2-transitive permutation groups.

ASL — affine special linear

Translations extended by SL instead of GL: ASL(n,𝔽q) = 𝔽qn ⋊ SL(n,𝔽q).

• Order qⁿ · |SL(n,𝔽q)| = qⁿ · |GL(n,𝔽q)| / (q − 1).
• Properties index q − 1 in AGL(n,𝔽q); equal to it when q = 2.

Semilinear groups

A semilinear map is x ↦ A · xσ, a linear map twisted by a field automorphism σ (a power of the Frobenius x ↦ xp). Since Gal(𝔽q / 𝔽p) is cyclic of order d = logp q, each semilinear group is the corresponding linear group extended by Cd — and equals it over a prime field (d = 1).

ΓL — general semilinear

ΓL(n,𝔽q) = GL(n,𝔽q) ⋊ Gal(𝔽q / 𝔽p).

• Order d · |GL(n,𝔽q)|, with d = logp q.
• Properties ΓL(n,𝔽p) = GL(n,𝔽p); the Frobenius acts on GL(n,𝔽q) by raising every matrix entry to the p-th power.

ΣL — special semilinear

ΣL(n,𝔽q) = SL(n,𝔽q) ⋊ Gal(𝔽q / 𝔽p), the semilinear analogue of SL.

• Order d · |SL(n,𝔽q)|.

PΓL — projective general semilinear

ΓL(n,𝔽q) modulo scalars — for n ≥ 3 the full automorphism group of the projective space PG(n−1, q) as an incidence geometry (Fundamental Theorem of Projective Geometry; a projective line carries no incidence structure, so the theorem starts in dimension 2).

• Order d · |PGL(n,𝔽q)|.
• Automorphisms Aut(PSL(2,𝔽q)) = PΓL(2,𝔽q) for q > 3; in dimension n ≥ 3 one more factor, the transpose-inverse graph automorphism, joins it.

PΣL — projective special semilinear

ΣL(n,𝔽q) modulo scalars; PSL(n,𝔽q) ⊴ PΣL(n,𝔽q) with cyclic quotient of order d.

• Connection PΓL / PΣL supply the outer automorphisms of PSL(n,𝔽q) coming from field automorphisms.

AΓL — affine general semilinear

AΓL(n,𝔽q) = 𝔽qn ⋊ ΓL(n,𝔽q) — affine maps allowing a field twist.

• Order qⁿ · d · |GL(n,𝔽q)|.
• Examples AΓL(1,𝔽4) ≅ S₄ (order 24); AΓL(1,𝔽8) is a solvable 2-transitive group of order 168.
• Connection AΓL(1,𝔽q) is 2-transitive on 𝔽q; its transitive subgroups account for the 1-dimensional affine 2-transitive groups (AGL(1,𝔽q) = 𝔽q ⋊ 𝔽q×, the d = 1 case, is the sharply 2-transitive Frobenius one).

AΣL — affine special semilinear

AΣL(n,𝔽q) = 𝔽qn ⋊ ΣL(n,𝔽q), the special-linear affine semilinear group.

• Order qⁿ · d · |SL(n,𝔽q)|.

Triangular groups

UT — unitriangular

Upper-triangular n × n matrices with 1s on the diagonal over 𝔽q.

• Order qn(n−1)/2 — a Sylow p-subgroup of GL(n,𝔽q) when q is a power of p.
• Examples UT(2,𝔽q) ≅ (𝔽q, +); UT(3,𝔽q) is the Heisenberg group H(3,𝔽q) (for q = 2, the dihedral group of order 8; for an odd prime q, the exponent-q extraspecial group).
• Properties nilpotent of class n − 1; every finite p-group embeds in some UT(n,𝔽p).

H — Heisenberg

Unitriangular matrices whose only free entries are the first row and the last column — m = 2k + 1 of them in a (k+2) × (k+2) matrix.

• Order |H(m, R)| = |R|m. The name's first argument is the group's dimension, not the matrix's, so H(5,𝔽4) is a 4 × 4 matrix of order 4⁵ = 1024.
• Nilpotent of class 2 — always, unlike UT(n,𝔽q), whose class grows with n. Center = derived subgroup = the corner entry ≅ (R, +).
• H(3, R) = UT(3, R) — in a 3 × 3 matrix the two shapes coincide; above that H is a proper subgroup of UT.
• Over a prime field H(2k+1,𝔽p) is the extraspecial group p+1+2k — the plus type only; the minus type is never Heisenberg.

Borel

The full upper-triangular subgroup of GL(n,𝔽q) — a diagonal torus extended by the unitriangular group: B(n,𝔽q) = T ⋉ UT(n,𝔽q), with T ≅ (Cq−1)ⁿ.

• Order (q − 1)ⁿ · qn(n−1)/2.
• Properties solvable and self-normalizing; the subgroups of GL(n,𝔽q) containing it are exactly the block-upper-triangular (parabolic) subgroups; its structure torus ⋉ unipotent is the model case of a Borel subgroup of an algebraic group.

Wikipedia: Group of Lie type