# Testing Finiteness In this section, $K$ is a finite degree extension of the field $F(x_1, \ldots, x_m)$, where $F$ is $Q$, a number field, or a finite field. Also $m \geq 0$ if char $F = 0$, and $m > 0$ otherwise. ## `IsFinite(G : parameters): GrpMat -> BoolElt, RngIntElt` ```magma NumberRandom : RngIntElt Default: 10 Presentation : MonStgElt Default: "CT" Small : RngIntElt Default: 10^6 OrderLimit : RngIntElt Default: 10^15 Algebra : BoolElt Default: true Nilpotent : BoolElt Default: false UseCongruence : BoolElt Default: false DetermineOrder: BoolElt Default: false Prime : RngintElt Default: 3 ``` Let $G$ be a finitely generated subgroup of ${\operatorname{GL}}(n, K)$. If $G$ is finite then the function returns `true`, otherwise `false`. The function is an implementation of algorithms from [[Detinko and Flannery, 2008](../../references.md#cite-draft), [Detinko and Flannery, 2009](../../references.md#cite-zero), [Detinko *et al.*, 2009](../../references.md#cite-positive), [Detinko *et al.*, 2013](../../references.md#cite-recog)]. The algorithm first tests whether `NumberRandom` random elements of $G$ have finite order. If the optional parameter `Algebra` is `true` and $K$ is a function field of characteristic zero (resp. positive characteristic), then we use the “algebra algorithm” of [[Detinko and Flannery, 2009](../../references.md#cite-zero)] (resp. [[Detinko *et al.*, 2009](../../references.md#cite-positive)]) to decide finiteness. Otherwise, we prove that $G$ is finite by first constructing a congruence homomorphism, then a presentation for the congruence image, and finally evaluates its relations to obtain normal generators for the congruence kernel. If char $K = 0$, then the kernel should be trivial, otherwise the kernel is unipotent. The optional parameter `Presentation` is used to dictate how the presentation is constructed. If its value is “CT”, then we use the presentation provided by `CompositionTreeVerify`. If its value is “PC” and the image is soluble, then we use a PC-presentation provided by `LMGSolubleRadical`. If its value is “FP” then we use the presentation provided by `FPGroup` or `FPGroupStrong`. If the order of the congruence image is less than the value of the optional argument `Small`, then we use `FPGroup` to construct the presentation; if it is less than the value of the optional argument `OrderLimit`, then we use `FPGroupStrong` to construct the presentation; otherwise we use the presentation provided by `CompositionTreeVerify`. If $K$ is $Q$ or a number field and `UseCongruence` is `true`, then use congruence homomorphism machinery to decide; otherwise use default algorithm. If $G$ is known to be nilpotent then by setting the optional parameter `Nilpotent` to `true`, the function will call a special procedure for testing finiteness of nilpotent groups (see [[Detinko and Flannery, 2008](../../references.md#cite-draft), Section 4.3]). If the optional parameter `DetermineOrder` is set to `true`, and $G$ is finite, then the function returns the order of $G$. This may sometimes be more expensive than deciding finiteness. The optional parameter `Prime` applies if $K$ has characteristic 0: if `Prime` is positive, then it is a lower bound for the characteristic of the congruence image; if it is 0 then the function constructs a congruence image defined over a field of characteristic 0. ## `IsomorphicCopy(G : parameters): GrpMat -> BoolElt, GrpMat, HomGrp` ```magma Presentation : MonStgElt Default: "CT" Small : RngIntElt Default: 10^6 OrderLimit : RngIntElt Default: 10^15 Verify : BoolElt Default: false Algebra : BoolElt Default: false StartDegree : RngIntElt Default: 1 EndDegree : RngIntElt Default: 5 CompletelyReducible: BoolElt Default: false ``` The input is a finite subgroup $G$ of ${\operatorname{GL}}(n, K)$. If the function succeeds, then it returns `true` and an isomorphic copy of $G$ in ${\operatorname{GL}}(n,{\bf F}_{q})$ where $q$ is a prime power; otherwise it returns `false`. A description of the method used is in [[Detinko *et al.*, 2013](../../references.md#cite-recog), Section 4.3]. If char $K > 0$, then ${\bf F}_{q}$ has the same characteristic. Note that the function always succeeds if $K$ has zero characteristic. If the optional parameter `Algebra` is `true` and $K$ is a function field of characteristic zero (resp. positive characteristic), then we use the “algebra algorithm” of [[Detinko and Flannery, 2009](../../references.md#cite-zero)] (resp. [[Detinko *et al.*, 2009](../../references.md#cite-positive)]) to construct an isomorphic copy. Otherwise we prove that a congruence homomorphism is an isomorphism by constructing a presentation for the congruence image and evaluating its relations to obtain normal generators for the congruence kernel. The optional parameter `Presentation` is used to dictate how the presentation is constructed. If its value is “CT”, then we use the presentation provided by `CompositionTreeVerify`. If its value is “PC” and the image is soluble, then we use a PC-presentation provided by `LMGSolubleRadical`. If its value is “FP” then we use the presentation provided by `FPGroup` or `FPGroupStrong`. If the order of the congruence image is less than the value of the optional argument `Small`, then we use `FPGroup` to construct the presentation; if it is less than the value of the optional argument `OrderLimit`, then we use `FPGroupStrong` to construct the presentation; otherwise we use the presentation provided by `CompositionTreeVerify`. If the optional parameter `Verify` is set to `true` then we first check whether $G$ is finite. If the characteristic of the coefficient field $F$ is positive, then we investigate extensions of $F$ in the range `StartDegree` $\ldots$ `EndDegree`. If the optional parameter `CompletelyReducible` is set to `true` then we use a more efficient algorithm to construct the isomorphic copy. ## `Order(G : parameters): GrpMat -> RngIntElt` ```magma Verify : BoolElt Default: false UseCongruence: BoolElt Default: false ``` Given a finite subgroup $G$ of ${\operatorname{GL}}(n,K)$, the function returns the order of $G$ by applying `IsomorphicCopy` to $G$. If the optional parameter `Verify` is set to `true`, then we first check that $G$ is finite. If $K$ is $Q$ or a number field and `UseCongruence` is `true`, then use congruence homomorphism machinery to decide; otherwise use default algorithm.