Many of the most important finite groups are naturally defined as matrix groups acting on a vector space, rather than as permutations. Magma provides a large family of functions that construct the classical linear, unitary, symplectic, and orthogonal groups directly as permutation groups, by realizing their natural action on the affine or projective points of a vector space.
Each function in this family returns two values: a permutation group G corresponding to the action of a matrix group M on a vector space V, and an indexed set of the affine or projective points on which M acts, giving the correspondence between that set and the G-set of G.
Specifying Degree and Field
Most of these functions are parameterized by a degree and a coefficient field, which can be supplied in one of three equivalent forms: integers n and q giving the degree and the field GF(q) (or GF(q^2) for the unitary groups); an integer n together with an explicit finite field K; or a vector space V = K^n on which the matrix group naturally acts.
> G, S := PGL(3, 2);
> G;
Permutation group G acting on a set of cardinality 7
Order = 168 = 2^3 * 3 * 7
> S;
{@ (1 0 0), (0 1 0), (0 0 1), (1 1 0), (1 0 1), (0 1 1), (1 1 1) @}
Here S is the indexed set of one-dimensional subspaces of GF(2)^3 (represented by their generating vectors), and S[i] corresponds to the point moved by G to position i.
Affine Groups
AffineGeneralLinearGroup(n, q) (short form AGL) constructs AGL(n, q), the permutation group corresponding to the action of GL(n, q) on the affine points of the n-dimensional vector space V over K = GF(q). AffineSpecialLinearGroup (ASL) does the same for SL(n, q) acting on the same affine points. Related affine families extend the linear part by field automorphisms or scalars: AffineGammaLinearGroup (AGammaL), AffineSigmaLinearGroup (ASigmaL), AffineSymplecticGroup (ASp), and AffineSigmaSymplecticGroup (ASigmaSp).
> G, V := AGL(2, 3);
> G;
Permutation group G acting on a set of cardinality 9
Order = 432 = 2^4 * 3^3
The degree of G here is |V| = q^n, since AGL(n, q) acts on all of the n-dimensional vector space, not just the nonzero or projective points.
A more general constructor, AffineGroup(M), takes any matrix group M of degree d over a finite field F and builds the semidirect product V:M, where V = F^d is the natural M-module. The result is a standard permutation group of degree |V| = |F|^d, together with the correspondence between elements of V and the points of the G-set.
Projective Groups
The projective groups act on the one-dimensional subspaces of V rather than on all of its vectors, so their degree is (q^n - 1)/(q - 1) instead of q^n. ProjectiveGeneralLinearGroup(n, q) (PGL) constructs the action of GL(n, q) on these projective points, and ProjectiveSpecialLinearGroup(n, q) (PSL) does the same for SL(n, q); both require n >= 2 and q a prime power. As with the affine case, Gamma and Sigma variants (PGammaL, PSigmaL) extend the group by field automorphisms.
> G, S := PSL(2, 5);
> G;
Permutation group G acting on a set of cardinality 6
Order = 60 = 2^2 * 3 * 5
PSL(2, 5) is realized here as a permutation group of degree 6, since the projective line over GF(5) has (5^2-1)/(5-1) = 6 points; this recovers the classical isomorphism PSL(2,5) = A5 in its natural degree-6 action.
Unitary, Symplectic, and Orthogonal Families
The same pattern extends to the other classical group families. ProjectiveGeneralUnitaryGroup (PGU) and ProjectiveSpecialUnitaryGroup (PSU), along with their Gamma/Sigma extensions PGammaU and PSigmaU, realize the unitary groups acting on projective points of a Hermitian space over GF(q^2). ProjectiveSymplecticGroup (PSp) and ProjectiveSigmaSymplecticGroup (PSigmaSp) do the same for the symplectic groups. The orthogonal groups need a sign to distinguish the geometry: ProjectiveGeneralOrthogonalGroup (PGO) requires odd n >= 3, while PGOPlus and PGOMinus handle the two inequivalent types of even-dimensional orthogonal space with n >= 2. ProjectiveSpecialOrthogonalGroup (PSO, with Plus/Minus variants) and the simple orthogonal groups ProjectiveOmega, ProjectiveOmegaPlus, ProjectiveOmegaMinus (POmega, POmegaPlus, POmegaMinus) follow the same odd/even-with-sign convention.
The Suzuki Group
ProjectiveSuzukiGroup(q) (PSz) is a special case: the Suzuki simple group Sz(q) only exists for q = 2^(2n+1), and its natural module is always four-dimensional, so it is parameterized purely by the field — as an integer q, a field K = GF(q), or the vector space K^4. As with the other projective constructors, it returns the permutation group together with the indexed set of projective points giving the correspondence to the G-set.
> G, S := PSz(8);
> G;
Permutation group G acting on a set of cardinality 65
Order = 29120 = 2^6 * 5 * 13