MAGMA Computational Algebra System

Computing with nilpotent matrix groups

In this section K = F(x₁, ..., x_m) where F is Q, a number field, or a finite field, and m ≥0.

IsUnipotent(G) : GrpMat -> BoolElt

This function takes as input a finitely generated matrix group G over K, and tests whether G is unipotent i.e. whether it is conjugate in GL(n, K) to a group of upper unitriangular matrices. If G is unipotent then the function returns true, otherwise false.

IsCompletelyReducibleNilpotent(G) : GrpMat -> BoolElt

IsCompletelyReducibleNilpotent(G : parameters) : GrpMat -> BoolElt

    Verify: BoolElt                     Default: false

This function takes as input a finitely generated nilpotent matrix group G over K, and tests whether G is completely reducible. If so, it returns true, otherwise false. If Verify is true, then the function checks that G is nilpotent.

IsNilpotent(G) : GrpMat -> BoolElt

Let G be a finitely generated subgroup of GL(n, K). This function returns true if G is nilpotent; otherwise it returns false. If K is finite then the function is an implementation of the algorithm of [DF06]. If K is infinite then the function works along the lines of an algorithm in [DF08], and is based on the construction of a homomorphic image H of G via CongruenceImage.

SylowSystem(G) : GrpMat -> []

SylowSystem(G : parameters) : GrpMat -> []

    Verify: BoolElt                     Default: false

Given a nilpotent matrix group G over a finite field, this function constructs one Sylow p-subgroup for each prime p dividing |G| using the algorithm of [DF06]. If the optional parameter Verify is set to true, then we first verify that G is nilpotent.