In this section intrinsics for providing information about the global properties of an fp-group are described.
A very basic requirement is to determine whether a given fp-group G is finite or infinite. Unfortunately, this is a problem for which there is no general solution. Instead, an intrinsic Order is provided which attempts to prove that the group is finite and a second intrinsic IsInfiniteFPGroup is provided which attempts to prove that the group is infinite. If both fail to produce a result then no conclusion can be drawn.
We assume that the group is finite and try to prove that this assumption is true using coset enumeration and other techniques.
The strategy employed by the functions Order and FactoredOrder may involve trying to obtain information on certain subgroups of G. Whether or not an attempt is made to construct a presentation for a subgroup arising in the course of the computation by means of Reidemeister-Schreier rewriting, is controlled by three parameters:
var UseRewrite: BoolElt Default: true var MinIndex: RngIntElt Default: 10 var MaxIndex: RngIntElt Default: 1000 If UseRewrite is set to false, attempts to construct presentations for subgroups are not made. Otherwise, MinIndex and MaxIndex specify for subgroups of which index range Reidemeister-Schreier rewriting is done.
The following strategy is used for trying to determine the order of G.
Experienced users can control the behaviour of coset enumerations which may be invoked by the functions Order and FactoredOrder with a wide range of parameters. Both functions -- in addition to the parameters mentioned above -- accept the same parameters as the function CosetEnumerationProcess in Subsec. Interactive Coset Enumeration.
The intrinsic IsInfiniteFPGroup attempts to prove that an fp-group is infinite by finding a subgroup that has an infinite abelian section.
Given an fp-group G, this function attempts to determine the order of G or to prove that G is infinite. If a finite order can be computed, the function Order returns the order as a positive integer, whereas the function FactoredOrder returns a sequence of prime power factors. The function FactoredOrder reports an error in all other cases, whereas the function Order returns the object Infinity, if G can be shown to be infinite and returns a value of 0 if neither a finite value for the group order nor a proof for the infinity of G can be obtained. No conclusions can be drawn from a return value 0 of Order.In addition to the parameters controlling possibly invoked coset enumerations, there exist some other parameters controlling the strategy used by the functions Order and FactoredOrder. These parameters are described below.
> G<x, y> := FPGroup<x,y | x^8, y^7, (x*y)^2, (x^-1*y)^3>;
> G;
Finitely presented group G on 2 generators
Relations
x^8 = Id(G)
y^7 = Id(G)
(x * y)^2 = Id(G)
(x^-1 * y)^3 = Id(G)
> Order(G);
10752
Given a finitely-presented group G, the function attempts to establish that G is infinite. If it succeeds it returns true. Otherwise it returns false meaning that the tests it has made don't prove the group to be infinite. This does not imply that the group is likely to be finite. The code searches for epimorphisms φ of G and uses the Holt-Plesken cohomological criterion, as implemented in the intrinsic HasPositiveH1Dimension, to test for non-finiteness of the kernel of φ. The version in this release works best on groups G which are perfect or which have small abelian quotients. It will be extended to handle groups which have larger soluble quotients in a subsequent release.At present epimorphisms are sought using four techniques that are described elsewhere in this chapter: L2-quotient, Simple Quotient, Low Index Subgroups and Low Index Normal Subgroups. The intrinsic has corresponding parameters L2Quot, SimQuot, LISub and LINSub that allow the user to control the types of epimorphisms to be used. The default is for all four methods to be employed -- however, the user can turn any of them off by setting the appropriate parameter to false. A verbose flag InfGrp will provide some information about progress.
The application of the Holt-Plesken theorem to an epimorphism φ requires a knowledge of the QH modules, where H is the image of φ. A database containing such modules for ATLAS groups of degree up to 1000 is available as an optional download. The serious user of IsInfiniteFPGroup is strongly advised to have this database available as it can save a great deal of runtime.
> SetVerbose("InfGrp", 2);
> G<x,y> := FPGroup<x,y|x^2, y^3, (x*y)^7, (x,y)^12>;
> inf := IsInfiniteFPGroup(G);
>
L2Q SEARCH commencing:
L2Q SEARCH Considering L2Qs with degrees in the interval [1, 1000]
SIMPLE QUOTIENT Commence searching interval [60, 10000]:
Found epi onto PSL(2, 7) of order 168 and degree 8: 0.000 secs.
Found epi onto PSL(2, 13) of order 1092 and degree 14: 0.060 secs.
SIMPLE QUOTIENT Search of interval [60, 10000] unsuccessful: 2 epis, 0.610
secs.
NORMAL SUBGROUPS Search of indices [1, 1000] commencing:
NORMAL SUBGROUPS Search of [1, 1000] unsuccessful: 1 + 0 subgroups, 0.200
secs.
SUBGRP SEARCH of indices [1, 25] commencing:
SUBGRP SEARCH of [1, 25] was unsuccessful: 8 + 8 subgroups, 0.990 secs.
SIMPLE QUOTIENT Commence searching interval [20001, 100000]:
SIMPLE QUOTIENT Search of interval [20001, 100000] unsuccessful: 0 epis,
0.620 secs.
NORMAL SUBGROUPS Search of indices [1001, 10000] commencing:
NORMAL SUBGROUPS Search of [1001, 10000] unsuccessful: 0 + 0 subgroups,
0.960 secs.
SUBGRP SEARCH of indices [26, 50] commencing:
SUBGRP SEARCH of [26, 50] was unsuccessful: 8 + 8 subgroups, 1.460 secs.
SIMPLE QUOTIENT Commence searching interval [100001, 1000000]:
Found epi onto J_2 of order 604800 and degree 100: 1.300 secs.
For 4950-dim ex_sq of perm module, dim of H^1/Z is 1: 1.050 secs.
SIMPLE QUOTIENT Search of interval [100001, 1000000] successful: 1 epis,
2.430 secs.
> inf;
true
So the group G has the simple group J2 as quotient and the rational
representation corresponding to the exterior square of J2's degree 100
permutation module lifted to G has first cohomology group of dimension 1.
So G is infinite by an application of the Holt-Plesken theorem.
Given a finitely-presented group G and an epimorphism φ of G onto a permutation group H, this intrinsic returns true if some (Q)H-module, where (Q) is the field of the rationals, lifts to a (Q)G-module whose first cohomology group is non-trivial. By a result of Holt and Plesken this shows that G is infinite. The (Q)H-modules are obtained in three ways. The first consists of (Q)H-modules that are easily obtainable from the permutation representation of H. The second method is to systematically construct the (Q)H-modules of H while the third source is the Magma database of rational representations of small simple groups. Parameters allow the user to specify which of these methods are used (see below).The parameter ESLimit allows the user to place a limit on the dimension of exterior squares for which the Holt-Plesken test will be applied. If the parameter CmpIrrs is set to true then Magma will construct rational irreducible representations for H if they are not present in the database. If the parameter DBIrrs is set to true then Magma will use those irreducible modules for H that are available in the Rational Representations Database. Setting both CmpIrrs and DBIrrs to false specifies that the test should be applied only to the permutation module and its exterior square.
There are two signatures for this intrinsic where the second has a third argument needed by the intrinsic IsInfiniteFPGroup. Note that this version is only intended to be called internally. For all other uses the two-argument version should be used.
Print: RngIntElt Default: 0
involgens: BoolElt Default: false
This functions calculates the values for which the small cancellation conditions T(i), C(j) and C'(k) are satisfied by the presentation G. It returns i, j, k', where T(i), C(j), and C'(k) are satisfied for all k > k'.If the parameter involgens is set true then edges labelled by involutory generators x in a van Kampen diagram for G are regarded as undirected, and are not regarded as pieces of the relator x2.
> G := FPGroup< x,y | x^6, y^6, (x*y)^5, (x,y)^3 >; > SmallCancellationConditions(G); This presentation satisfies T( 3 ). This presentation satisfies C( 5 ). This presentation satisfies C'(k) for all k > 1/5 . 3 5 1/5
Attempt to show that G has a subgroup with homomorphic image the free group of rank 2. If successful return true, plus a subgroup witness to this property. If not successful, return false. Note that false does not imply that G doesn't have the stated property, just that no such subgroup has been found. The witness group returned has a finite index subgroup with the free group of rank 2 as a homomorphic image. The algorithm will consider subgroups of G with index h having L≤h≤U.We attempt to decide if G is large by calculating the (multivariable) Alexander polynomial of those subgroups H of G having index h in the given range, and also subgroups of H having index up to 2h in G.
Simplify: BoolElt Default: trueControls simplification of subgroup presentations.MaxExponent: RngIntElt Default: 100000If the abelian torsion of subgroup H is greater than MaxExponent we do not rewrite to obtain a presentation of H.TimeLimit: RngIntElt Default: 10If, in computing low index subgroups of a finite index subgroup H of G, the low-index subgroup calculation takes more than TimeLimit in seconds, then abort without trying to find more subgroups of H.