Properties of an FP-Group

In this section intrinsics for providing information about the global properties of an fp-group are described.

Cardinality

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.

(1)

Check whether G is free. If so, G is either trivial or infinite.

(2)

Check whether the presentation for G is deficient (i.e. whether the number of relations is smaller than the number of generators). If it is, G is infinite.

(3)

Check the subgroups of G with known order. If such a subgroup is known to be infinite or if we can compute its index in G, we're done.

(4)

Try to compute the index of G in a supergroup of known order. (An infinite supergroup in which G has finite index proves G to be infinite.)

(5)

Try to enumerate the cosets of the trivial subgroup in G.

(6)

Check the subgroups of known or easily computable index in G. If we can compute the order of such a subgroup or prove that it is infinite, we're done.

(7)

Try to enumerate the cosets of some subgroups occurring "naturally" in the presentation of G.

(8)

Check the supergroups in which G has known or easily computable index. If we can compute the order of a supergroup or prove that it is infinite, we're done.

(9)

Try to rewrite G w.r.t. some supergroup and to enumerate the cosets of the trivial subgroup using the resulting presentation.

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.

Order(G: parameters) : GrpFP -> RngIntElt

FactoredOrder(G: parameters) : GrpFP -> [ <RngIntElt, RngIntElt> ]

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

> 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

    Print: RngIntElt                    Default: 0

    involgens: BoolElt                  Default: false

> 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