Groups of Lie Type

The groups of Lie type are built from finite fields. A finite field F_q exists for every prime power q = pⁿ, and these are the only finite fields up to isomorphism. The simplest examples are Z/pZ for p prime, but when n > 1 the construction requires adjoining roots of irreducible polynomials over Z/pZ. Every finite field has a cyclic multiplicative group of order q − 1, and its structure is exactly the raw material needed to build matrix groups.

The general linear group GL(n, q) of invertible n × n matrices over F_q is not simple: its scalar matrices form a normal subgroup, and so do the matrices of determinant 1. Passing to the special linear group SL(n, q) and then dividing out its center gives the projective special linear group PSL(n, q). This group is simple for every n ≥ 2 and every q, except PSL(2, 2) ≅ S₃ and PSL(2, 3) ≅ A₄.

Every simple Lie algebra over the complex numbers is described by a Dynkin diagram: one node per simple root, joined by one, two or three lines according to the angle between the roots, with an arrow pointing toward the shorter root when two roots have different lengths. The connected diagrams are the four infinite series Aₙ, Bₙ, Cₙ, Dₙ and the five exceptions E₆, E₇, E₈, F₄ and G₂.

In 1955 Claude Chevalley showed that each diagram gives a group over every finite field. Aₙ gives PSL(n + 1, q), Bₙ the orthogonal group PΩ(2n + 1, q), Cₙ the symplectic group PSp(2n, q), Dₙ the orthogonal group PΩ⁺(2n, q), and the exceptional diagrams give G₂(q), F₄(q), E₆(q), E₇(q) and E₈(q). The order always has the same shape: q^N times a product of factors qᵈ − 1, divided by the size of a center. Here N is the number of positive roots and the exponents d are the degrees of the diagram’s Weyl group; the demo shows the formula for each choice.

Bₙ and Cₙ have the same order formula. In characteristic 2 they give the same group, but when q is odd and n ≥ 3 they are different groups of the same order, a coincidence Emil Artin noticed in 1955. Nine diagram types plus seven twisted versions make the 16 families of simple groups of Lie type.

A symmetry of a Dynkin diagram permutes the simple roots and so gives an automorphism of the Chevalley group. Combining it with a field automorphism of matching order and taking the fixed points gives a new group. Robert Steinberg did this in 1959 for the flips of Aₙ, Dₙ and E₆ and for the triality rotation of D₄, producing the unitary groups ²Aₙ(q) = PSU(n + 1, q), the orthogonal groups ²Dₙ(q) = PΩ⁻(2n, q), ²E₆(q) and ³D₄(q). Folding the diagram along the symmetry shows the shape of the new group’s own root system: Aₙ folds to Cₘ or BCₘ, Dₙ to Bₙ₋₁, E₆ to F₄, and D₄ under triality to G₂.

B₂, G₂ and F₄ look symmetric if you ignore the arrow, but turning them around swaps long and short roots. In characteristic 2 for B₂ and F₄, and characteristic 3 for G₂, the group has an extra automorphism that does exactly this. When q is an odd power of the characteristic, the field has an automorphism whose square is the Frobenius map, and the fixed points form the Suzuki groups ²B₂(q) (Michio Suzuki, 1960) and the Ree groups ²G₂(q) and ²F₄(q) (Rimhak Ree, 1960–61). The Suzuki groups are the only non-abelian simple groups whose order is not divisible by 3. The smallest case ²F₄(2) is not simple, but its index-2 subgroup, the Tits group ²F₄(2)′, is; it is sometimes counted as a 27th sporadic group.

For q ≥ 4, PSL(2, q) is a simple group of order q(q − 1)(q + 1)/gcd(2, q − 1). It acts on the projective line over F_q, which has q + 1 points, by Möbius transformations z ↦ (az + b)/(cz + d), and this action is doubly transitive. On a log-log plot the order is a line of slope 3, and the odd-q points sit a factor of 2 below the even ones because for odd q the center {±1} of SL(2, q) has order 2.

Several small cases coincide with other simple groups. PSL(2, 4) ≅ PSL(2, 5) ≅ A₅, the icosahedral group of order 60. PSL(2, 7) ≅ PSL(3, 2) = GL(3, 2) has 168 elements and is the symmetry group of the Fano plane below. PSL(2, 9) ≅ A₆. These exceptional isomorphisms are low-dimensional accidents that do not continue.

Dickson classified all subgroups of PSL(2, q) in 1901: cyclic and dihedral groups, A₄, S₄, A₅, the upper triangular Borel subgroups, and smaller groups of the same kind. Arguments in the classification often reduce to this case by studying involutions and their centralizers.

The Fano plane is the projective plane over F₂: its 7 points are the nonzero vectors of F₂³ and its 7 lines are the 2-dimensional subspaces, each holding 3 points. Every invertible 3 × 3 matrix over F₂ sends subspaces to subspaces, so it is a symmetry of the plane, and every symmetry arises this way. There are 7 × 6 × 4 = 168 such matrices: the first column can be any nonzero vector, the second any vector other than 0 and the first, the third anything outside the plane they span.

Over F₂ the only scalar is 1, so GL(3, 2) = SL(3, 2) = PSL(3, 2). That this group of order 168 is isomorphic to PSL(2, 7), which permutes the 8 points of the projective line over F₇, is one of the exceptional isomorphisms: the two descriptions look nothing alike. It is also the symmetry group of Klein’s quartic curve, whose 168 conformal symmetries make it a famous object in geometry.