MAGMA Computational Algebra System

Construction of Congruence Homomorphisms and Testing Finiteness

In this section, K is the field F(x₁, ..., x_m), where F is Z, Q, a number field, or a finite field. Also m ≥0 if char F = 0, and m > 0 otherwise.

CongruenceImage(G) : GrpMat -> GrpMat

Let G be a finitely generated subgroup of GL(n, K). If m = 0 (respectively, m > 0) then the function constructs a homomorphic image of G in GL(n, q) for some prime power q (respectively, in GL(n, F)). If char K = p > 0 then torsion elements of the kernel of a related congruence homomorphism are p-elements; otherwise the kernel is torsion-free. If m > 0 and char K = 0, then by applying the function CongruenceImage twice, one obtains a congruence image of G over a finite field. For a detailed description of the congruence homomorphisms see [DF08, Section 3]. The function returns the congruence image, and the list of images of generators of G.

IsomorphicCopy(G) : GrpMat -> BoolElt, GrpMat

IsomorphicCopy(G : parameters) : GrpMat -> BoolElt, GrpMat

    Verify: BoolElt                     Default: false

    CompletelyReducible: BoolElt        Default: false

    Nilpotent: BoolElt                  Default: false

    StartDegree: RngIntElt              Default: 1

    EndDegree: RngIntElt                Default: 5

    Limit: RngIntElt                    Default: 0

The input is a finite subgroup G of GL(n, K).

If the function succeeds, then it returns true and an isomorphic copy of G in GL(n, q) where q is a prime power; otherwise it returns false.

If the optional parameter Verify is set to true then we first check whether G is finite.

If G is nilpotent or completely reducible, a more efficient algorithm is invoked by setting the corresponding optional parameter to true.

If the characteristic of the coefficient field F is positive, then we investigate extensions of F in the range StartDegree ... EndDegree. If Limit is positive, consider Limit random elements from F, otherwise consider all elements of F.

IsFinite(G) : GrpMat -> BoolElt

IsFinite(G : parameters) : GrpMat -> BoolElt

    CompletelyReducible: BoolElt        Default: false

    Nilpotent: BoolElt                  Default: false

Let G be a finitely generated subgroup of GL(n, K). If G is finite then the function returns true, otherwise false. If char K = 0 (respectively, char K > 0) and m > 0, then the function is an implementation of the algorithm of [DF09] (respectively, [DFO]). If G is known to be nilpotent then by setting the optional parameter to true, the function will call a special procedure for testing finiteness of nilpotent groups (see [DF08, Section 4.3] if char K = 0, and [DFO, Section 4] if char K > 0). If char K > 0 and G is known to be completely reducible, then one can set the optional parameter to true to call a simpler procedure ([DFO, Section 2]). If char K = 0, m = 0, and G is not nilpotent, then this function calls IsFinite.

Order(G) : GrpMat -> RngIntElt

Order(G : parameters) : GrpMat -> RngIntElt

    Verify: BoolElt                     Default: false

    Nilpotent: BoolElt                  Default: false

Given a finite subgroup G of GL(n, K), the function returns the order of G by applying IsomorphicCopy to G. If G is known to be nilpotent then we can set the optional parameter to true, so that the function will call a special procedure for computing the order of a nilpotent group over a finite field. To verify finiteness of G one sets the optional parameter to true.