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