Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 Bib Ind
[Top of Book] [Contents] [Previous Chapter] [Next Chapter]

[MathJax on]

47 Finitely Presented Groups
47.1 IsSubgroupFpGroup and IsFpGroup
47.1-1 IsSubgroupFpGroup
47.1-2 IsFpGroup
47.1-3 InfoFpGroup
47.2 Creating Finitely Presented Groups
47.2-1 \/
47.2-2 FactorGroupFpGroupByRels
47.2-3 ParseRelators
47.2-4 StringFactorizationWord
47.3 Comparison of Elements of Finitely Presented Groups
47.3-1 \=
47.3-2 \<
47.3-3 FpElmComparisonMethod
47.3-4 SetReducedMultiplication
47.4 Preimages in the Free Group
47.4-1 FreeGroupOfFpGroup
47.4-2 FreeGeneratorsOfFpGroup
47.4-3 RelatorsOfFpGroup
47.4-4 UnderlyingElement
47.4-5 ElementOfFpGroup
47.5 Operations for Finitely Presented Groups
47.5-1 PseudoRandom
47.6 Coset Tables and Coset Enumeration
47.6-1 CosetTable
47.6-2 TracedCosetFpGroup
47.6-3 FactorCosetAction
47.6-4 CosetTableBySubgroup
47.6-5 CosetTableFromGensAndRels
47.6-6 CosetTableDefaultMaxLimit
47.6-7 CosetTableDefaultLimit
47.6-8 MostFrequentGeneratorFpGroup
47.6-9 IndicesInvolutaryGenerators
47.7 Standardization of coset tables
47.7-1 CosetTableStandard
47.7-2 StandardizeTable
47.8 Coset tables for subgroups in the whole group
47.8-1 CosetTableInWholeGroup
47.8-2 SubgroupOfWholeGroupByCosetTable
47.9 Augmented Coset Tables and Rewriting
47.9-1 AugmentedCosetTableInWholeGroup
47.9-2 AugmentedCosetTableMtc
47.9-3 AugmentedCosetTableRrs
47.9-4 RewriteWord
47.10 Low Index Subgroups
47.10-1 LowIndexSubgroupsFpGroupIterator
47.11 Converting Groups to Finitely Presented Groups
47.11-1 IsomorphismFpGroup
47.11-2 IsomorphismFpGroupByGenerators
47.12 New Presentations and Presentations for Subgroups
47.12-1 IsomorphismSimplifiedFpGroup
47.13 Preimages under Homomorphisms from an FpGroup
47.13-1 SubgroupOfWholeGroupByQuotientSubgroup
47.13-2 IsSubgroupOfWholeGroupByQuotientRep
47.13-3 AsSubgroupOfWholeGroupByQuotient
47.13-4 DefiningQuotientHomomorphism
47.14 Quotient Methods
47.14-1 PQuotient
47.14-2 EpimorphismQuotientSystem
47.14-3 EpimorphismPGroup
47.14-4 EpimorphismNilpotentQuotient
47.14-5 SolvableQuotient
47.14-6 EpimorphismSolvableQuotient
47.14-7 LargerQuotientBySubgroupAbelianization
47.15 Abelian Invariants for Subgroups
47.15-1 AbelianInvariantsSubgroupFpGroup
47.15-2 AbelianInvariantsSubgroupFpGroupMtc
47.15-3 AbelianInvariantsSubgroupFpGroupRrs
47.15-4 AbelianInvariantsNormalClosureFpGroup
47.15-5 AbelianInvariantsNormalClosureFpGroupRrs
47.16 Testing Finiteness of Finitely Presented Groups
47.16-1 IsInfiniteAbelianizationGroup
47.16-2 NewmanInfinityCriterion

### 47 Finitely Presented Groups

A finitely presented group (in short: FpGroup) is a group generated by a finite set of abstract generators subject to a finite set of relations that these generators satisfy. Every finite group can be represented as a finitely presented group, though in almost all cases it is computationally much more efficient to work in another representation (even the regular permutation representation).

Finitely presented groups are obtained by factoring a free group by a set of relators. Their elements know about this presentation and compare accordingly.

So to create a finitely presented group you first have to generate a free group (see `FreeGroup` (37.2-1) for details). Then a list of relators is constructed as words in the generators of the free group and is factored out to obtain the finitely presented group. Its generators are the images of the free generators. So for example to create the group

⟨ a, b ∣ a^2, b^3, (a b)^5 ⟩

you can use the following commands:

    gap> f := FreeGroup( "a", "b" );;
    gap> g := f / [ f.1^2, f.2^3, (f.1*f.2)^5 ];
    <fp group on the generators [ a, b ]>

Note that you cannot call the generators by their names. These names are not variables, but just display figures. So, if you want to access the generators by their names, you first have to introduce the respective variables and to assign the generators to them.

    gap> Unbind(a);
    gap> GeneratorsOfGroup( g );
    [ a, b ]
    gap> a;
    Error, Variable: 'a' must have a value
    gap> a := g.1;; b := g.2;; # assign variables
    gap> GeneratorsOfGroup( g );
    [ a, b ]
    gap> a in f;
    false
    gap> a in g;
    true

To relieve you of the tedium of typing the above assignments, when working interactively, there is the function `AssignGeneratorVariables` (37.2-3).

Note that the generators of the free group are different from the generators of the FpGroup (even though they are displayed by the same names). That means that words in the generators of the free group are not elements of the finitely presented group. Vice versa elements of the FpGroup are not words.

    gap> a*b = b*a;
    false
    gap> (b^2*a*b)^2 = a^0;
    true

Such calculations comparing elements of an FpGroup may run into problems: There exist finitely presented groups for which no algorithm exists (it is known that no such algorithm can exist) that will tell for two arbitrary words in the generators whether the corresponding elements in the FpGroup are equal.

Therefore the methods used by GAP to compute in finitely presented groups may run into warning errors, run out of memory or run forever. If the FpGroup is (by theory) known to be finite the algorithms are guaranteed to terminate (if there is sufficient memory available), but the time needed for the calculation cannot be bounded a priori. See 47.6 and 47.16.

    gap> (b^2*a*b)^2;
    (b^2*a*b)^2
    gap> a^0;
    <identity ...>

A consequence of our convention is that elements of finitely presented groups are not printed in a unique way. See also `SetReducedMultiplication` (47.3-4).

#### 47.1 IsSubgroupFpGroup and IsFpGroup

##### 47.1-1 IsSubgroupFpGroup

`‣ IsSubgroupFpGroup`( H ) | ( category )

returns `true` if H is a finitely presented group or a subgroup of a finitely presented group.

##### 47.1-2 IsFpGroup

`‣ IsFpGroup`( G ) | ( filter )

is a synonym for `IsSubgroupFpGroup(G) and IsGroupOfFamily(G)`.

Free groups are a special case of finitely presented groups, namely finitely presented groups with no relators.

Another special case are groups given by polycyclic presentations. GAP uses a special representation for these groups which is created in a different way. See chapter 46 for details.

    gap> g:=FreeGroup(2);
    <free group on the generators [ f1, f2 ]>
    gap> IsFpGroup(g);
    true
    gap> h:=CyclicGroup(2);
    <pc group of size 2 with 1 generators>
    gap> IsFpGroup(h);
    false

##### 47.1-3 InfoFpGroup

`‣ InfoFpGroup` | ( info class )

The info class for functions dealing with finitely presented groups is `InfoFpGroup`.

#### 47.2 Creating Finitely Presented Groups

##### 47.2-1 \/

`‣ \/`( F, rels ) | ( method )

creates a finitely presented group given by the presentation ⟨ gens ∣ rels ⟩ where gens are the free generators of the free group F. Note that relations are entered as relators, i.e., as words in the generators of the free group. To enter an equation use the quotient operator, i.e., for the relation a^b = ab one has to enter a^b / (a b).

The same result is obtained with the infix operator `/`, i.e., as F `/` rels.

    gap> f := FreeGroup( 3 );;
    gap> f / [ f.1^4, f.2^3, f.3^5, f.1*f.2*f.3 ];
    <fp group on the generators [ f1, f2, f3 ]>

##### 47.2-2 FactorGroupFpGroupByRels

`‣ FactorGroupFpGroupByRels`( G, elts ) | ( function )

returns the factor group G/N of G by the normal closure N of elts where elts is expected to be a list of elements of G.

##### 47.2-3 ParseRelators

`‣ ParseRelators`( gens, rels ) | ( function )

Will translate a list of relations as given in print, e.g. x y^2 = (x y^3 x)^2 xy = yzx into relators. gens must be a list of generators of a free group, each being displayed by a single letter. rels is a string that lists a sequence of equalities. These must be written in the letters which are the names of the generators in gens. Change of upper/lower case is interpreted to indicate inverses.

    gap> f:=FreeGroup("x","y","z");;
    gap> AssignGeneratorVariables(f);
    #I  Assigned the global variables [ x, y, z ]
    gap> r:=ParseRelators([x,y,z],
    > "x^2 = y^5 = z^3 = (xyxyxy^4)^2 = (xz)^2 = (y^2z)^2 = 1");
    [ x^2, y^5, z^3, (x*z)^2, (y^2*z)^2, ((x*y)^3*y^3)^2 ]
    gap> g:=f/r;
    <fp group on the generators [ x, y, z ]>

##### 47.2-4 StringFactorizationWord

`‣ StringFactorizationWord`( w ) | ( function )

returns a string that expresses a given word w in compact form written as a string. Inverses are expressed by changing the upper/lower case of the generators, recurring expressions are written as products.

    gap> StringFactorizationWord(z^-1*x*y*y*y*x*x*y*y*y*x*y^-1*x);
    "Z(xy3x)2Yx"

#### 47.3 Comparison of Elements of Finitely Presented Groups

##### 47.3-1 \=

`‣ \=`( a, b ) | ( method )

Two elements of a finitely presented group are equal if they are equal in this group. Nevertheless they may be represented as different words in the generators. Because of the fundamental problems mentioned in the introduction to this chapter such a test may take very long and cannot be guaranteed to finish.

The method employed by GAP for such an equality test use the underlying finitely presented group. First (unless this group is known to be infinite) GAP tries to find a faithful permutation representation by a bounded Todd-Coxeter. If this fails, a Knuth-Bendix (see 52.6) is attempted and the words are compared via their normal form.

If only elements in a subgroup are to be tested for equality it thus can be useful to translate the problem in a new finitely presented group by rewriting (see `IsomorphismFpGroup` (47.11-1));

The equality test of elements underlies many "basic" calculations, such as the order of an element, and the same type of problems can arise there. In some cases, working with rewriting systems can still help to solve the problem. The kbmag package provides such functionality, see the package manual for further details.

##### 47.3-2 \<

`‣ \<`( a, b ) | ( method )

Compared with equality testing, problems get even worse when trying to compute a total ordering on the elements of a finitely presented group. As any ordering that is guaranteed to be reproducible in different runs of GAP or even with different groups given by syntactically equal presentations would be prohibitively expensive to implement, the ordering of elements is depending on a method chosen by GAP and not guaranteed to stay the same when repeating the construction of an FpGroup. The only guarantee given for the `<` ordering for such elements is that it will stay the same for one family during its lifetime. The attribute `FpElmComparisonMethod` (47.3-3) is used to obtain a comparison function for a family of FpGroup elements.

##### 47.3-3 FpElmComparisonMethod

`‣ FpElmComparisonMethod`( fam ) | ( attribute )

If fam is the elements family of a finitely presented group this attribute returns a function `smaller(left, right)` that will be used to compare elements in fam.

##### 47.3-4 SetReducedMultiplication

`‣ SetReducedMultiplication`( obj ) | ( function )

For an FpGroup obj, an element obj of it or the family obj of its elements, this function will force immediate reduction when multiplying, keeping words short at extra cost per multiplication.

#### 47.4 Preimages in the Free Group

##### 47.4-1 FreeGroupOfFpGroup

`‣ FreeGroupOfFpGroup`( G ) | ( attribute )

returns the underlying free group for the finitely presented group G. This is the group generated by the free generators provided by the `FreeGeneratorsOfFpGroup` (47.4-2) value of G.

##### 47.4-2 FreeGeneratorsOfFpGroup

`‣ FreeGeneratorsOfFpGroup`( G ) | ( attribute )

`‣ FreeGeneratorsOfWholeGroup`( U ) | ( operation )

`FreeGeneratorsOfFpGroup` returns the underlying free generators corresponding to the generators of the finitely presented group G which must be a full FpGroup.

`FreeGeneratorsOfWholeGroup` also works for subgroups of an FpGroup and returns the free generators of the full group that defines the family.

##### 47.4-3 RelatorsOfFpGroup

`‣ RelatorsOfFpGroup`( G ) | ( attribute )

returns the relators of the finitely presented group G as words in the free generators provided by the `FreeGeneratorsOfFpGroup` (47.4-2) value of G.

    gap> f := FreeGroup( "a", "b" );;
    gap> g := f / [ f.1^5, f.2^2, f.1^f.2*f.1 ];
    <fp group on the generators [ a, b ]>
    gap> Size( g );
    10
    gap> FreeGroupOfFpGroup( g ) = f;
    true
    gap> FreeGeneratorsOfFpGroup( g );
    [ a, b ]
    gap> RelatorsOfFpGroup( g );
    [ a^5, b^2, b^-1*a*b*a ]

Note that these attributes are only available for the full finitely presented group. It is possible (for example by using `Subgroup` (39.3-1)) to construct a subgroup of index 1 which is not identical to the whole group. The latter one can be obtained in this situation via `Parent` (31.7-1).

Elements of a finitely presented group are not words, but are represented using a word from the free group as representative. The following two commands obtain this representative, respectively create an element in the finitely presented group.

##### 47.4-4 UnderlyingElement

`‣ UnderlyingElement`( elm ) | ( operation )

Let elm be an element of a group whose elements are represented as words with further properties. Then `UnderlyingElement` returns the word from the free group that is used as a representative for elm.

    gap> w := g.1*g.2;
    a*b
    gap> IsWord( w );
    false
    gap> ue := UnderlyingElement( w );
    a*b
    gap> IsWord( ue );
    true

##### 47.4-5 ElementOfFpGroup

`‣ ElementOfFpGroup`( fam, word ) | ( operation )

If fam is the elements family of a finitely presented group and word is a word in the free generators underlying this finitely presented group, this operation creates the element with the representative word in the free group.

    gap> ge := ElementOfFpGroup( FamilyObj( g.1 ), f.1*f.2 );
    a*b
    gap> ge in f;
    false
    gap> ge in g;
    true

#### 47.5 Operations for Finitely Presented Groups

Finitely presented groups are groups and so all operations for groups should be applicable to them (though not necessarily efficient methods are available). Most methods for finitely presented groups rely on coset enumeration. See 47.6 for details.

The command `IsomorphismPermGroup` (43.3-1) can be used to obtain a faithful permutation representation, if such a representation of small degree exists. (Otherwise it might run very long or fail.)

    gap> f := FreeGroup( "a", "b" );
    <free group on the generators [ a, b ]>
    gap> g := f / [ f.1^2, f.2^3, (f.1*f.2)^5 ];
    <fp group on the generators [ a, b ]>
    gap> h := IsomorphismPermGroup( g );
    [ a, b ] -> [ (1,2)(4,5), (2,3,4) ]
    gap> u:=Subgroup(g,[g.1*g.2]);;rt:=RightTransversal(g,u);
    RightTransversal(<fp group of size 60 on the generators
    [ a, b ]>,Group([ a*b ]))
    gap> Image(ActionHomomorphism(g,rt,OnRight));
    Group([ (1,2)(3,4)(5,7)(6,8)(9,10)(11,12),
      (1,3,2)(4,5,6)(7,8,9)(10,11,12) ])

##### 47.5-1 PseudoRandom

`‣ PseudoRandom`( F: radius := l ) | ( method )

The default algorithm for `PseudoRandom` (30.7-2) makes little sense for finitely presented or free groups, as it produces words that are extremely long.

By specifying the option `radius`, instead elements are taken as words in the generators of F in the ball of radius l with equal distribution in the free group.

    gap> PseudoRandom(g:radius:=20);
    a^3*b^2*a^-2*b^-1*a*b^-4*a*b^-1*a*b^-4

#### 47.6 Coset Tables and Coset Enumeration

Coset enumeration (see [Neu82] for an explanation) is one of the fundamental tools for the examination of finitely presented groups. This section describes GAP functions that can be used to invoke a coset enumeration.

Note that in addition to the built-in coset enumerator there is the GAP package ACE. Moreover, GAP provides an interactive Todd-Coxeter in the GAP package ITC which is based on the XGAP package.

##### 47.6-1 CosetTable

`‣ CosetTable`( G, H ) | ( operation )

returns the coset table of the finitely presented group G on the cosets of the subgroup H.

Basically a coset table is the permutation representation of the finitely presented group on the cosets of a subgroup (which need not be faithful if the subgroup has a nontrivial core). Most of the set theoretic and group functions use the regular representation of G, i.e., the coset table of G over the trivial subgroup.

The coset table is returned as a list of lists. For each generator of G and its inverse the table contains a generator list. A generator list is simply a list of integers. If l is the generator list for the generator g and if l[i] = j then generator g takes the coset i to the coset j by multiplication from the right. Thus the permutation representation of G on the cosets of H is obtained by applying `PermList` (42.5-2) to each generator list.

The coset table is standard (see below).

For finitely presented groups, a coset table is computed by a Todd-Coxeter coset enumeration. Note that you may influence the performance of that enumeration by changing the values of the global variables `CosetTableDefaultLimit` (47.6-7) and `CosetTableDefaultMaxLimit` (47.6-6) described below and that the options described under `CosetTableFromGensAndRels` (47.6-5) are recognized.

    gap> tab := CosetTable(g, Subgroup(g, [ g.1, g.2*g.1*g.2*g.1*g.2^-1 ]));
    [ [ 1, 4, 5, 2, 3 ], [ 1, 4, 5, 2, 3 ], [ 2, 3, 1, 4, 5 ],
      [ 3, 1, 2, 4, 5 ] ]
    gap> List( last, PermList );
    [ (2,4)(3,5), (2,4)(3,5), (1,2,3), (1,3,2) ]
    gap> PrintArray( TransposedMat( tab ) );
    [ [  1,  1,  2,  3 ],
      [  4,  4,  3,  1 ],
      [  5,  5,  1,  2 ],
      [  2,  2,  4,  4 ],
      [  3,  3,  5,  5 ] ]

The last printout in the preceding example provides the coset table in the form in which it is usually used in hand calculations: The rows correspond to the cosets, the columns correspond to the generators and their inverses in the ordering g_1, g_1^{-1}, g_2, g_2^{-1}. (See section 47.7 for a description on the way the numbers are assigned.)

##### 47.6-2 TracedCosetFpGroup

`‣ TracedCosetFpGroup`( tab, word, pt ) | ( function )

Traces the coset number pt under the word word through the coset table tab. (Note: word must be in the free group, use `UnderlyingElement` (47.4-4) if in doubt.)

    gap> TracedCosetFpGroup(tab,UnderlyingElement(g.1),2);
    4

##### 47.6-3 FactorCosetAction

`‣ FactorCosetAction`( G, H ) | ( operation )

returns the action of G on the cosets of its subgroup H.

    gap> u := Subgroup( g, [ g.1, g.1^g.2 ] );
    Group([ a, b^-1*a*b ])
    gap> FactorCosetAction( g, u );
    [ a, b ] -> [ (2,4)(5,6), (1,2,3)(4,5,6) ]

##### 47.6-4 CosetTableBySubgroup

`‣ CosetTableBySubgroup`( G, H ) | ( operation )

returns a coset table for the action of G on the cosets of H. The columns of the table correspond to the `GeneratorsOfGroup` (39.2-4) value of G.

##### 47.6-5 CosetTableFromGensAndRels

`‣ CosetTableFromGensAndRels`( fgens, grels, fsgens ) | ( function )

is an internal function which is called by the functions `CosetTable` (47.6-1), `CosetTableInWholeGroup` (47.8-1) and others. It is, in fact, the proper working horse that performs a Todd-Coxeter coset enumeration. fgens must be a set of free generators and grels a set of relators in these generators. fsgens are subgroup generators expressed as words in these generators. The function returns a coset table with respect to fgens.

`CosetTableFromGensAndRels` will call `TCENUM.CosetTableFromGensAndRels`. This makes it possible to replace the built-in coset enumerator with another one by assigning `TCENUM` to another record.

The library version which is used by default performs a standard Felsch strategy coset enumeration. You can call this function explicitly as `GAPTCENUM.CosetTableFromGensAndRels` even if other coset enumerators are installed.

The expected parameters are

fgens

  generators of the free group F

grels

  relators as words in F

fsgens

  subgroup generators as words in F.

`CosetTableFromGensAndRels` processes two options (see chapter 8):

`max`

  The limit of the number of cosets to be defined. If the enumeration does not finish with this number of cosets, an error is raised and the user is asked whether she wants to continue. The default value is the value given in the variable `CosetTableDefaultMaxLimit`. (Due to the algorithm the actual limit used can be a bit higher than the number given.)

`silent`

  If set to `true` the algorithm will not raise the error mentioned under option `max` but silently return `fail`. This can be useful if an enumeration is only wanted unless it becomes too big.

##### 47.6-6 CosetTableDefaultMaxLimit

`‣ CosetTableDefaultMaxLimit` | ( global variable )

is the default limit for the number of cosets allowed in a coset enumeration.

A coset enumeration will not finish if the subgroup does not have finite index, and even if it has it may take many more intermediate cosets than the actual index of the subgroup is. To avoid a coset enumeration "running away" therefore GAP has a "safety stop" built in. This is controlled by the global variable `CosetTableDefaultMaxLimit`.

If this number of cosets is reached, GAP will issue an error message and prompt the user to either continue the calculation or to stop it. The default value is 4096000.

See also the description of the options to `CosetTableFromGensAndRels` (47.6-5).

    gap> f := FreeGroup( "a", "b" );;
    gap> u := Subgroup( f, [ f.2 ] );
    Group([ b ])
    gap> Index( f, u );
    Error, the coset enumeration has defined more than 4096000 cosets
     called from
    TCENUM.CosetTableFromGensAndRels( fgens, grels, fsgens ) called from
    CosetTableFromGensAndRels( fgens, grels, fsgens ) called from
    TryCosetTableInWholeGroup( H ) called from
    CosetTableInWholeGroup( H ) called from
    IndexInWholeGroup( H ) called from
    ...
    Entering break read-eval-print loop ...
    type 'return;' if you want to continue with a new limit of 8192000 cosets,
    type 'quit;' if you want to quit the coset enumeration,
    type 'maxlimit := 0; return;' in order to continue without a limit
    brk> quit;

At this point, a `break`-loop (see Section 6.4) has been entered. The line beginning `Error` tells you why this occurred. The next seven lines occur if `OnBreak` (6.4-3) has its default value `Where` (6.4-5). They explain, in this case, how GAP came to be doing a coset enumeration. Then you are given a number of options of how to escape the `break`-loop: you can either continue the calculation with a larger number of permitted cosets, stop the calculation if you don't expect the enumeration to finish (like in the example above), or continue without a limit on the number of cosets. (Choosing the first option will, of course, land you back in a `break`-loop. Try it!)

Setting `CosetTableDefaultMaxLimit` (or the `max` option value, for any function that invokes a coset enumeration) to `infinity` (18.2-1) (or to 0) will force all coset enumerations to continue until they either get a result or exhaust the whole available space. For example, each of the following two inputs

    gap> CosetTableDefaultMaxLimit := 0;;
    gap> Index( f, u );

or

    gap> Index( f, u : max := 0 );

have essentially the same effect as choosing the third option (typing: `maxlimit := 0; return;`) at the `brk>` prompt above (instead of `quit;`).

##### 47.6-7 CosetTableDefaultLimit

`‣ CosetTableDefaultLimit` | ( global variable )

is the default number of cosets with which any coset table is initialized before doing a coset enumeration.

The function performing this coset enumeration will automatically extend the table whenever necessary (as long as the number of cosets does not exceed the value of `CosetTableDefaultMaxLimit` (47.6-6)), but this is an expensive operation. Thus, if you change the value of `CosetTableDefaultLimit`, you should set it to a number of cosets that you expect to be sufficient for your subsequent coset enumerations. On the other hand, if you make it too large, your job will unnecessarily waste a lot of space.

The default value of `CosetTableDefaultLimit` is 1000.

##### 47.6-8 MostFrequentGeneratorFpGroup

`‣ MostFrequentGeneratorFpGroup`( G ) | ( function )

is an internal function which is used in some applications of coset table methods. It returns the first of those generators of the given finitely presented group G which occur most frequently in the relators.

##### 47.6-9 IndicesInvolutaryGenerators

`‣ IndicesInvolutaryGenerators`( G ) | ( attribute )

returns the indices of those generators of the finitely presented group G which are known to be involutions. This knowledge is used by internal functions to improve the performance of coset enumerations.

#### 47.7 Standardization of coset tables

For any two coset numbers i and j with i < j the first occurrence of i in a coset table precedes the first occurrence of j with respect to the usual row-wise ordering of the table entries. Following the notation of Charles Sims' book on computation with finitely presented groups [Sim94] we call such a table a standard coset table.

The table entries which contain the first occurrences of the coset numbers i > 1 recursively provide for each i a representative of the corresponding coset in form of a unique word w_i in the generators and inverse generators of G. The first coset (which is H itself) can be represented by the empty word w_1. A coset table is standard if and only if the words w_1, w_2, ... are length-plus-lexicographic ordered (as defined in [Sim94]), for short: lenlex.

This standardization of coset tables is different from that used in GAP versions 4.2 and earlier. Before that, we ignored the columns that correspond to inverse generators and hence only considered words in the generators of G. We call this older ordering the semilenlex standard as it also applies to the case of semigroups where no inverses of the generators are known.

We changed our default from the semilenlex standard to the lenlex standard to be consistent with [Sim94]. However, the semilenlex standardisation remains available and the convention used for all implicit standardisations can be selected by setting the value of the global variable `CosetTableStandard` (47.7-1) to either `"lenlex"` or `"semilenlex"`. Independent of the current value of `CosetTableStandard` (47.7-1) you can standardize (or restandardize) a coset table at any time using `StandardizeTable` (47.7-2).

##### 47.7-1 CosetTableStandard

`‣ CosetTableStandard` | ( global variable )

specifies the definition of a standard coset table. It is used whenever coset tables or augmented coset tables are created. Its value may be `"lenlex"` or `"semilenlex"`. If it is `"lenlex"` coset tables will be standardized using all their columns as defined in Charles Sims' book (this is the new default standard of GAP). If it is `"semilenlex"` they will be standardized using only their generator columns (this was the original GAP standard). The default value of `CosetTableStandard` is `"lenlex"`.

##### 47.7-2 StandardizeTable

`‣ StandardizeTable`( table, standard ) | ( function )

standardizes the given coset table table. The second argument is optional. It defines the standard to be used, its values may be `"lenlex"` or `"semilenlex"` specifying the new or the old convention, respectively. If no value for the parameter standard is provided the function will use the global variable `CosetTableStandard` (47.7-1) instead. Note that the function alters the given table, it does not create a copy.

    gap> StandardizeTable( tab, "semilenlex" );
    gap> PrintArray( TransposedMat( tab ) );
    [ [  1,  1,  2,  4 ],
      [  3,  3,  4,  1 ],
      [  2,  2,  3,  3 ],
      [  5,  5,  1,  2 ],
      [  4,  4,  5,  5 ] ]

#### 47.8 Coset tables for subgroups in the whole group

##### 47.8-1 CosetTableInWholeGroup

`‣ CosetTableInWholeGroup`( H ) | ( attribute )

`‣ TryCosetTableInWholeGroup`( H ) | ( operation )

is equivalent to `CosetTable(G,H)` where G is the (unique) finitely presented group such that H is a subgroup of G. It overrides a `silent` option (see `CosetTableFromGensAndRels` (47.6-5)) with `false`.

The variant `TryCosetTableInWholeGroup` does not override the `silent` option with `false` in case a coset table is only wanted if not too expensive. It will store a result that is not `fail` in the attribute `CosetTableInWholeGroup`.

##### 47.8-2 SubgroupOfWholeGroupByCosetTable

`‣ SubgroupOfWholeGroupByCosetTable`( fpfam, tab ) | ( function )

takes a family fpfam of an FpGroup and a coset table tab and returns the subgroup of fpfam`!.wholeGroup` defined by this coset table. See also `CosetTableBySubgroup` (47.6-4).

#### 47.9 Augmented Coset Tables and Rewriting

##### 47.9-1 AugmentedCosetTableInWholeGroup

`‣ AugmentedCosetTableInWholeGroup`( H[, gens] ) | ( function )

For a subgroup H of a finitely presented group, this function returns an augmented coset table. If a generator set gens is given, it is guaranteed that gens will be a subset of the primary and secondary subgroup generators of this coset table.

It is mutable so we are permitted to add further entries. However existing entries may not be changed. Any entries added however should correspond to the subgroup only and not to an homomorphism.

##### 47.9-2 AugmentedCosetTableMtc

`‣ AugmentedCosetTableMtc`( G, H, type, string ) | ( function )

is an internal function used by the subgroup presentation functions described in 48.2. It applies a Modified Todd-Coxeter coset representative enumeration to construct an augmented coset table (see 48.2) for the given subgroup H of G. The subgroup generators will be named string `1`, string `2`, ....

The function accepts the options `max` and `silent` as described for the function `CosetTableFromGensAndRels` (47.6-5).

##### 47.9-3 AugmentedCosetTableRrs

`‣ AugmentedCosetTableRrs`( G, table, type, string ) | ( function )

is an internal function used by the subgroup presentation functions described in 48.2. It applies the Reduced Reidemeister-Schreier method to construct an augmented coset table for the subgroup of G which is defined by the given coset table table. The new subgroup generators will be named string `1`, string `2`, ....

##### 47.9-4 RewriteWord

`‣ RewriteWord`( aug, word ) | ( function )

`RewriteWord` rewrites word (which must be a word in the underlying free group with respect to which the augmented coset table aug is given) in the subgroup generators given by the augmented coset table aug. It returns a Tietze-type word (i.e. a list of integers), referring to the primary and secondary generators of aug.

If word is not contained in the subgroup, `fail` is returned.

#### 47.10 Low Index Subgroups

##### 47.10-1 LowIndexSubgroupsFpGroupIterator

`‣ LowIndexSubgroupsFpGroupIterator`( G[, H], index[, excluded] ) | ( operation )

`‣ LowIndexSubgroupsFpGroup`( G[, H], index[, excluded] ) | ( operation )

These functions compute representatives of the conjugacy classes of subgroups of the finitely presented group G that contain the subgroup H of G and that have index less than or equal to index.

`LowIndexSubgroupsFpGroupIterator` returns an iterator (see 30.8) that can be used to run over these subgroups, and `LowIndexSubgroupsFpGroup` returns the list of these subgroups. If one is interested only in one or a few subgroups up to a given index then preferably the iterator should be used.

If the optional argument excluded has been specified, then it is expected to be a list of words in the free generators of the underlying free group of G, and `LowIndexSubgroupsFpGroup` returns only those subgroups of index at most index that contain H, but do not contain any conjugate of any of the group elements defined by these words.

If not given, H defaults to the trivial subgroup.

The algorithm used finds the requested subgroups by systematically running through a tree of all potential coset tables of G of length at most index (where it skips all branches of that tree for which it knows in advance that they cannot provide new classes of such subgroups). The time required to do this depends, of course, on the presentation of G, but in general it will grow exponentially with the value of index. So you should be careful with the choice of index.

    gap> li:=LowIndexSubgroupsFpGroup( g, TrivialSubgroup( g ), 10 );
    [ Group(<fp, no generators known>), Group(<fp, no generators known>),
      Group(<fp, no generators known>), Group(<fp, no generators known>) ]

By default, the algorithm computes no generating sets for the subgroups. This can be enforced with `GeneratorsOfGroup` (39.2-4):

    gap> GeneratorsOfGroup(li[2]);
    [ a, b*a*b^-1 ]

If we are interested just in one (proper) subgroup of index at most 10, we can use the function that returns an iterator. The first subgroup found is the group itself, except if a list of excluded elements is entered (see below), so we look at the second subgroup.

    gap> iter:= LowIndexSubgroupsFpGroupIterator( g, 10 );;
    gap> s1:= NextIterator( iter );;  Index( g, s1 );
    1
    gap> IsDoneIterator( iter );
    false
    gap> s2:= NextIterator( iter );;  s2 = li[2];
    true

As an example for an application of the optional parameter excluded, we compute all conjugacy classes of torsion free subgroups of index at most 24 in the group G = ⟨ x,y,z ∣ x^2, y^4, z^3, (xy)^3, (yz)^2, (xz)^3 ⟩. It is know from theory that each torsion element of this group is conjugate to a power of x, y, z, xy, xz, or yz. (Note that this includes conjugates of y^2.)

    gap> F := FreeGroup( "x", "y", "z" );;
    gap> x := F.1;; y := F.2;; z := F.3;;
    gap> G := F / [ x^2, y^4, z^3, (x*y)^3, (y*z)^2, (x*z)^3 ];;
    gap> torsion := [ x, y, y^2, z, x*y, x*z, y*z ];;
    gap> SetInfoLevel( InfoFpGroup, 2 );
    gap> lis := LowIndexSubgroupsFpGroup(G, TrivialSubgroup(G), 24, torsion);;
    #I  LowIndexSubgroupsFpGroup called
    #I   class 1 of index 24 and length 8
    #I   class 2 of index 24 and length 24
    #I   class 3 of index 24 and length 24
    #I   class 4 of index 24 and length 24
    #I   class 5 of index 24 and length 24
    #I  LowIndexSubgroupsFpGroup done. Found 5 classes
    gap> SetInfoLevel( InfoFpGroup, 0 );

If a particular image group is desired, the operation `GQuotients` (40.9-4) (see 47.14) can be useful as well.

#### 47.11 Converting Groups to Finitely Presented Groups

##### 47.11-1 IsomorphismFpGroup

`‣ IsomorphismFpGroup`( G ) | ( attribute )

returns an isomorphism from the given finite group G to a finitely presented group isomorphic to G. The function first chooses a set of generators of G and then computes a presentation in terms of these generators.

    gap> g := Group( (2,3,4,5), (1,2,5) );;
    gap> iso := IsomorphismFpGroup( g );
    [ (4,5), (1,2,3,4,5), (1,3,2,4,5) ] -> [ F1, F2, F3 ]
    gap> fp := Image( iso );
    <fp group on the generators [ F1, F2, F3 ]>
    gap> RelatorsOfFpGroup( fp );
    [ F1^2, F1^-1*F2*F1*F2^-1*F3*F2^-2, F1^-1*F3*F1*F2*F3^-1*F2*F3*F2^-1,
      F2^5*F3^-5, F2^5*(F3^-1*F2^-1)^2, (F2^-2*F3^2)^2 ]

##### 47.11-2 IsomorphismFpGroupByGenerators

`‣ IsomorphismFpGroupByGenerators`( G, gens[, string] ) | ( function )

`‣ IsomorphismFpGroupByGeneratorsNC`( G, gens, string ) | ( operation )

returns an isomorphism from a finite group G to a finitely presented group F isomorphic to G. The generators of F correspond to the generators of G given in the list gens. If string is given it is used to name the generators of the finitely presented group.

The `NC` version will avoid testing whether the elements in gens generate G.

    gap> SetInfoLevel( InfoFpGroup, 1 );
    gap> iso := IsomorphismFpGroupByGenerators( g, [ (1,2), (1,2,3,4,5) ] );
    #I  the image group has 2 gens and 5 rels of total length 39
    [ (1,2), (1,2,3,4,5) ] -> [ F1, F2 ]
    gap> fp := Image( iso );
    <fp group of size 120 on the generators [ F1, F2 ]>
    gap> RelatorsOfFpGroup( fp );
    [ F1^2, F2^5, (F2^-1*F1)^4, (F2*F1*F2^-1*F1)^3, (F2*F1*F2^-2*F1*F2)^2
     ]

The main task of the function `IsomorphismFpGroupByGenerators` is to find a presentation of G in the provided generators gens. In the case of a permutation group G it does this by first constructing a stabilizer chain of G and then it works through that chain from the bottom to the top, recursively computing a presentation for each of the involved stabilizers. The method used is essentially an implementation of John Cannon's multi-stage relations-finding algorithm as described in [Neu82] (see also [Can73] for a more graph theoretical description). Moreover, it makes heavy use of Tietze transformations in each stage to avoid an explosion of the total length of the relators.

Note that because of the random methods involved in the construction of the stabilizer chain the resulting presentations of G will in general be different for repeated calls with the same arguments.

    gap> M12 := MathieuGroup( 12 );
    Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6),
      (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ])
    gap> gens := GeneratorsOfGroup( M12 );;
    gap> iso := IsomorphismFpGroupByGenerators( M12, gens );;
    #I  the image group has 3 gens and 20 rels of total length 497
    gap> iso := IsomorphismFpGroupByGenerators( M12, gens );;
    #I  the image group has 3 gens and 19 rels of total length 493

Also in the case of a permutation group G, the function `IsomorphismFpGroupByGenerators` supports the option `method` that can be used to modify the strategy. The option `method` may take the following values.

`method := "regular"`

  This may be specified for groups of small size, up to 10^5 say. It implies that the function first constructs a regular representation R of G and then a presentation of R. In general, this presentation will be much more concise than the default one, but the price is the time needed for the construction of R.

`method := [ "regular", bound ]`

  This is a refinement of the previous possibility. In this case, `bound` should be an integer, and if so the method `"regular"` as described above is applied to the largest stabilizer in the stabilizer chain of G whose size does not exceed the given bound and then the multi-stage algorithm is used to work through the chain from that subgroup to the top.

`method := "fast"`

  This chooses an alternative method which essentially is a kind of multi-stage algorithm for a stabilizer chain of G but does not make any attempt do reduce the number of relators as it is done in Cannon's algorithm or to reduce their total length. Hence it is often much faster than the default method, but the total length of the resulting presentation may be huge.

`method := "default"`

  This simply means that the default method shall be used, which is the case if the option `method` is not given a value.

    gap> iso := IsomorphismFpGroupByGenerators( M12, gens :
    >                                           method := "regular" );;
    #I  the image group has 3 gens and 11 rels of total length 92
    gap> iso := IsomorphismFpGroupByGenerators( M12, gens :
    >                                           method := "fast" );;
    #I  the image group has 3 gens and 151 rels of total length 3658

Though the option `method := "regular"` is only checked in the case of a permutation group it also affects the performance and the results of the function `IsomorphismFpGroupByGenerators` for other groups, e. g. for matrix groups. This happens because, for these groups, the function first calls the function `NiceMonomorphism` (40.5-2) to get a bijective action homomorphism from G to a suitable permutation group, P say, and then, recursively, calls itself for the group P so that now the option becomes relevant.

    gap> G := ImfMatrixGroup( 5, 1, 3 );
    ImfMatrixGroup(5,1,3)
    gap> gens := GeneratorsOfGroup( G );
    [ [ [ -1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 0 ],
          [ -1, -1, -1, -1, 2 ], [ -1, 0, 0, 0, 1 ] ],
      [ [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ],
          [ 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1 ] ] ]
    gap> iso := IsomorphismFpGroupByGenerators( G, gens );;
    #I  the image group has 2 gens and 11 rels of total length 150
    gap> iso := IsomorphismFpGroupByGenerators( G, gens :
    >                                           method := "regular");;
    #I  the image group has 2 gens and 6 rels of total length 56
    gap> SetInfoLevel( InfoFpGroup, 0 );
    gap> iso;
    <composed isomorphism:[ [ [ -1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, \
    0, 0, 1, 0 ], [ -1, -1, -1, -1, 2 ], [ -1, 0, 0, 0, 1 ] ], [ [ 0, 1, 0\
    , 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 1, 0, 0, 0, 0 ], [ 0\
    , 0, 0, 0, 1 ] ] ]->[ F1, F2 ]>
    gap> ConstituentsCompositionMapping(iso);
    [ <action isomorphism>,
      [ (2,3,4)(5,6)(8,9,10), (1,2,3,5)(6,7,8,9) ] -> [ F1, F2 ] ]

Since GAP cannot decompose elements of a matrix group into generators, the resulting isomorphism is stored as a composition of a (faithful) permutation action on vectors and a homomorphism from the permutation image to the finitely presented group. In such a situation the constituent mappings can be obtained via `ConstituentsCompositionMapping` (32.2-7) as separate GAP objects.

#### 47.12 New Presentations and Presentations for Subgroups

`IsomorphismFpGroup` (47.11-1) is also used to compute a new finitely presented group that is isomorphic to the given subgroup of a finitely presented group. (This is typically the only method to compute with subgroups of a finitely presented group.)

    gap> f:=FreeGroup(2);;
    gap> g:=f/[f.1^2,f.2^3,(f.1*f.2)^5];
    <fp group on the generators [ f1, f2 ]>
    gap> u:=Subgroup(g,[g.1*g.2]);
    Group([ f1*f2 ])
    gap> hom:=IsomorphismFpGroup(u);
    [ <[ [ 1, 1 ] ]|f2^-1*f1^-1> ] -> [ F1 ]
    gap> new:=Range(hom);
    <fp group on the generators [ F1 ]>
    gap> List(GeneratorsOfGroup(new),i->PreImagesRepresentative(hom,i));
    [ <[ [ 1, 1 ] ]|f2^-1*f1^-1> ]

When working with such homomorphisms, some subgroup elements are expressed as extremely long words in the group generators. Therefore the underlying words of subgroup generators stored in the isomorphism (as obtained by `MappingGeneratorsImages` (40.10-2) and displayed when `View` (6.3-3)ing the homomorphism) as well as preimages under the homomorphism are stored in the form of straight line program elements (see 37.9). These will behave like ordinary words and no extra treatment should be necessary.

    gap> r:=Range(hom).1^10;
    F1^10
    gap> p:=PreImagesRepresentative(hom,r);
    <[ [ 1, 10 ] ]|(f2^-1*f1^-1)^10>

If desired, it also is possible to convert these underlying words using `EvalStraightLineProgElm` (37.9-4):

    gap> r:=EvalStraightLineProgElm(UnderlyingElement(p));
    (f2^-1*f1^-1)^10
    gap> p:=ElementOfFpGroup(FamilyObj(p),r);
    (f2^-1*f1^-1)^10

(If you are only interested in a finitely presented group isomorphic to the given subgroup but not in the isomorphism, you may also use the functions `PresentationViaCosetTable` (48.1-5) and `FpGroupPresentation` (48.1-4) (see 48.1).)

Homomorphisms can also be used to obtain an isomorphic finitely presented group with a (hopefully) simpler presentation.

##### 47.12-1 IsomorphismSimplifiedFpGroup

`‣ IsomorphismSimplifiedFpGroup`( G ) | ( attribute )

applies Tietze transformations to a copy of the presentation of the given finitely presented group G in order to reduce it with respect to the number of generators, the number of relators, and the relator lengths.

The operation returns an isomorphism with source G, range a group H isomorphic to G, so that the presentation of H has been simplified using Tietze transformations.

    gap> f:=FreeGroup(3);;
    gap> g:=f/[f.1^2,f.2^3,(f.1*f.2)^5,f.1/f.3];
    <fp group on the generators [ f1, f2, f3 ]>
    gap> hom:=IsomorphismSimplifiedFpGroup(g);
    [ f1, f2, f3 ] -> [ f1, f2, f1 ]
    gap> Range(hom);
    <fp group on the generators [ f1, f2 ]>
    gap> RelatorsOfFpGroup(Range(hom));
    [ f1^2, f2^3, (f1*f2)^5 ]
    gap> RelatorsOfFpGroup(g);
    [ f1^2, f2^3, (f1*f2)^5, f1*f3^-1 ]

`IsomorphismSimplifiedFpGroup` uses Tietze transformations to simplify the presentation, see 48.1-6.

#### 47.13 Preimages under Homomorphisms from an FpGroup

For some subgroups of a finitely presented group the number of subgroup generators increases with the index of the subgroup. However often these generators are not needed at all for further calculations, but what is needed is the action of the cosets of the subgroup. This gives the image of the subgroup in a finite quotient and this finite quotient can be used to calculate normalizers, closures, intersections and so forth [Hul01].

The same applies for subgroups that are obtained as preimages under homomorphisms.

##### 47.13-1 SubgroupOfWholeGroupByQuotientSubgroup

`‣ SubgroupOfWholeGroupByQuotientSubgroup`( fpfam, Q, U ) | ( function )

takes a FpGroup family fpfam, a finitely generated group Q such that the fp generators of fpfam can be mapped by an epimorphism phi onto the `GeneratorsOfGroup` (39.2-4) value of Q, and a subgroup U of Q. It returns the subgroup of fpfam`!.wholeGroup` which is the full preimage of U under phi.

##### 47.13-2 IsSubgroupOfWholeGroupByQuotientRep

`‣ IsSubgroupOfWholeGroupByQuotientRep`( G ) | ( representation )

is the representation for subgroups of an FpGroup, given by a quotient subgroup. The components G`!.quot` and G`!.sub` hold quotient, respectively subgroup.

##### 47.13-3 AsSubgroupOfWholeGroupByQuotient

`‣ AsSubgroupOfWholeGroupByQuotient`( U ) | ( attribute )

returns the same subgroup in the representation `AsSubgroupOfWholeGroupByQuotient`.

See also `SubgroupOfWholeGroupByCosetTable` (47.8-2) and `CosetTableBySubgroup` (47.6-4).

This technique is used by GAP for example to represent the derived subgroup, which is obtained from the quotient G/G'.

    gap> f:=FreeGroup(2);;g:=f/[f.1^6,f.2^6,(f.1*f.2)^6];;
    gap> d:=DerivedSubgroup(g);
    Group(<fp, no generators known>)
    gap> Index(g,d);
    36

##### 47.13-4 DefiningQuotientHomomorphism

`‣ DefiningQuotientHomomorphism`( U ) | ( function )

if U is a subgroup in quotient representation (`IsSubgroupOfWholeGroupByQuotientRep` (47.13-2)), this function returns the defining homomorphism from the whole group to U`!.quot`.

#### 47.14 Quotient Methods

An important class of algorithms for finitely presented groups are the quotient algorithms which compute quotient groups of a given finitely presented group. There are algorithms for epimorphisms onto abelian groups, p-groups and solvable groups. (The "low index" algorithm –`LowIndexSubgroupsFpGroup` (47.10-1)– can be considered as well as an algorithm that produces permutation group quotients.)

`MaximalAbelianQuotient` (39.18-4), as defined for general groups, returns the largest abelian quotient of the given group.

    gap> f:=FreeGroup(2);;fp:=f/[f.1^6,f.2^6,(f.1*f.2)^12];
    <fp group on the generators [ f1, f2 ]>
    gap> hom:=MaximalAbelianQuotient(fp);
    [ f1, f2 ] -> [ f1, f3 ]
    gap> Size(Image(hom));
    36

##### 47.14-1 PQuotient

`‣ PQuotient`( F, p[, c][, logord][, ctype] ) | ( function )

computes a factor p-group of a finitely presented group F in form of a quotient system. The quotient system can be converted into an epimorphism from F onto the p-group computed by the function `EpimorphismQuotientSystem` (47.14-2).

For a group G define the exponent-p central series of G inductively by cal P_1(G) = G and cal P_{i+1}(G) = [cal P_i(G),G]cal P_{i+1}(G)^p. The factor groups modulo the terms of the lower exponent-p central series are p-groups. The group G has p-class c if cal P_c(G) ≠ cal P_{c+1}(G) = 1.

The algorithm computes successive quotients modulo the terms of the exponent-p central series of F. If the parameter c is present, then the factor group modulo the (c+1)-th term of the exponent-p central series of F is returned. If c is not present, then the algorithm attempts to compute the largest factor p-group of F. In case F does not have a largest factor p-group, the algorithm will not terminate.

By default the algorithm computes only with factor groups of order at most p^256. If the parameter logord is present, it will compute with factor groups of order at most p^logord. If this parameter is specified, then the parameter c must also be given. The present implementation produces an error message if the order of a p-quotient exceeds p^256 or p^logord, respectively. Note that the order of intermediate p-groups may be larger than the final order of a p-quotient.

The parameter ctype determines the type of collector that is used for computations within the factor p-group. ctype must either be `"single"` in which case a simple collector from the left is used or `"combinatorial"` in which case a combinatorial collector from the left is used.

##### 47.14-2 EpimorphismQuotientSystem

`‣ EpimorphismQuotientSystem`( quotsys ) | ( operation )

For a quotient system quotsys obtained from the function `PQuotient` (47.14-1), this operation returns an epimorphism F → P where F is the finitely presented group of which quotsys is a quotient system and P is a pc group isomorphic to the quotient of F determined by quotsys.

Different calls to this operation will create different groups P, each with its own family.

    gap> PQuotient( FreeGroup(2), 5, 10, 1024, "combinatorial" );
    <5-quotient system of 5-class 10 with 520 generators>
    gap> phi := EpimorphismQuotientSystem( last );
    [ f1, f2 ] -> [ a1, a2 ]
    gap> Collected( Factors( Size( Image( phi ) ) ) );
    [ [ 5, 520 ] ]

##### 47.14-3 EpimorphismPGroup

`‣ EpimorphismPGroup`( fpgrp, p[, cl] ) | ( operation )

computes an epimorphism from the finitely presented group fpgrp to the largest p-group of p-class cl which is a quotient of fpgrp. If cl is omitted, the largest finite p-group quotient (of p-class up to 1000) is determined.

    gap> hom:=EpimorphismPGroup(fp,2);
    [ f1, f2 ] -> [ a1, a2 ]
    gap> Size(Image(hom));
    8
    gap> hom:=EpimorphismPGroup(fp,3,7);
    [ f1, f2 ] -> [ a1, a2 ]
    gap> Size(Image(hom));
    6561

##### 47.14-4 EpimorphismNilpotentQuotient

`‣ EpimorphismNilpotentQuotient`( fpgrp[, n] ) | ( function )

returns an epimorphism on the class n finite nilpotent quotient of the finitely presented group fpgrp. If n is omitted, the largest finite nilpotent quotient (of p-class up to 1000) is taken.

    gap> hom:=EpimorphismNilpotentQuotient(fp,7);
    [ f1, f2 ] -> [ f1*f4, f2*f5 ]
    gap> Size(Image(hom));
    52488

A related operation which is also applicable to finitely presented groups is `GQuotients` (40.9-4), which computes all epimorphisms from a (finitely presented) group F onto a given (finite) group G.

    gap> GQuotients(fp,Group((1,2,3),(1,2)));
    [ [ f1, f2 ] -> [ (1,2), (2,3) ], [ f1, f2 ] -> [ (2,3), (1,2,3) ],
      [ f1, f2 ] -> [ (1,2,3), (2,3) ] ]

##### 47.14-5 SolvableQuotient

`‣ SolvableQuotient`( F, size ) | ( function )

`‣ SolvableQuotient`( F, primes ) | ( function )

`‣ SolvableQuotient`( F, tuples ) | ( function )

`‣ SQ`( F, ... ) | ( function )

This routine calls the solvable quotient algorithm for a finitely presented group F. The quotient to be found can be specified in the following ways: Specifying an integer size finds a quotient of size up to size (if such large quotients exist). Specifying a list of primes in primes finds the largest quotient involving the given primes. Finally tuples can be used to prescribe a chief series.

`SQ` can be used as a synonym for `SolvableQuotient`.

##### 47.14-6 EpimorphismSolvableQuotient

`‣ EpimorphismSolvableQuotient`( F, param ) | ( function )

computes an epimorphism from the finitely presented group fpgrp to the largest solvable quotient given by param (specified as in `SolvableQuotient` (47.14-5)).

    gap> f := FreeGroup( "a", "b", "c", "d" );;
    gap> fp := f / [ f.1^2, f.2^2, f.3^2, f.4^2, f.1*f.2*f.1*f.2*f.1*f.2,
    >  f.2*f.3*f.2*f.3*f.2*f.3*f.2*f.3, f.3*f.4*f.3*f.4*f.3*f.4,
    > f.1^-1*f.3^-1*f.1*f.3, f.1^-1*f.4^-1*f.1*f.4,
    > f.2^-1*f.4^-1*f.2*f.4 ];;
    gap> hom:=EpimorphismSolvableQuotient(fp,300);Size(Image(hom));
    [ a, b, c, d ] -> [ f1*f2, f1*f2, f2*f3, f2 ]
    12
    gap> hom:=EpimorphismSolvableQuotient(fp,[2,3]);Size(Image(hom));
    [ a, b, c, d ] -> [ f1*f2*f4, f1*f2*f6*f8, f2*f3, f2 ]
    1152

##### 47.14-7 LargerQuotientBySubgroupAbelianization

`‣ LargerQuotientBySubgroupAbelianization`( hom, U ) | ( function )

Let hom a homomorphism from a finitely presented group G to a finite group H and Ule H. This function will -- if it exists -- return a subgroup SleG, such that the core of S is properly contained in the kernel of hom as well as in V', where V is the pre-image of U under hom. Thus S exposes a larger quotient of G. If no such subgroup exists, fail is returned.

    gap> f:=FreeGroup("x","y","z");;
    gap> g:=f/ParseRelators(f,"x^3=y^3=z^5=(xyx^2y^2)^2=(xz)^2=(yz^3)^2=1");
    <fp group on the generators [ x, y, z ]>
    gap> l:=LowIndexSubgroupsFpGroup(g,6);;
    gap> List(l,IndexInWholeGroup);
    [ 1, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6 ]
    gap> q:=DefiningQuotientHomomorphism(l[6]);;p:=Image(q);Size(p);
    Group([ (4,5,6), (1,2,3)(4,6,5), (2,4,6,3,5) ])
    360
    gap> s:=LargerQuotientBySubgroupAbelianization(q,SylowSubgroup(p,3));
    Group(<fp, no generators known>)
    gap> Size(Image(DefiningQuotientHomomorphism(s)));
    193273528320

#### 47.15 Abelian Invariants for Subgroups

Using variations of coset enumeration it is possible to compute the abelian invariants of a subgroup of a finitely presented group without computing a complete presentation for the subgroup in the first place. Typically, the operation `AbelianInvariants` (39.16-1) when called for subgroups should automatically take care of this, but in case you want to have further control about the methods used, the following operations might be of use.

##### 47.15-1 AbelianInvariantsSubgroupFpGroup

`‣ AbelianInvariantsSubgroupFpGroup`( G, H ) | ( function )

`AbelianInvariantsSubgroupFpGroup` is a synonym for `AbelianInvariantsSubgroupFpGroupRrs` (47.15-3).

##### 47.15-2 AbelianInvariantsSubgroupFpGroupMtc

`‣ AbelianInvariantsSubgroupFpGroupMtc`( G, H ) | ( function )

uses the Modified Todd-Coxeter method to compute the abelian invariants of a subgroup H of a finitely presented group G.

##### 47.15-3 AbelianInvariantsSubgroupFpGroupRrs

`‣ AbelianInvariantsSubgroupFpGroupRrs`( G, H ) | ( function )

`‣ AbelianInvariantsSubgroupFpGroupRrs`( G, table ) | ( function )

uses the Reduced Reidemeister-Schreier method to compute the abelian invariants of a subgroup H of a finitely presented group G.

Alternatively to the subgroup H, its coset table table in G may be given as second argument.

##### 47.15-4 AbelianInvariantsNormalClosureFpGroup

`‣ AbelianInvariantsNormalClosureFpGroup`( G, H ) | ( function )

`AbelianInvariantsNormalClosureFpGroup` is a synonym for `AbelianInvariantsNormalClosureFpGroupRrs` (47.15-5).

##### 47.15-5 AbelianInvariantsNormalClosureFpGroupRrs

`‣ AbelianInvariantsNormalClosureFpGroupRrs`( G, H ) | ( function )

uses the Reduced Reidemeister-Schreier method to compute the abelian invariants of the normal closure of a subgroup H of a finitely presented group G. See 48.2 for details on the different strategies.

The following example shows a calculation for the Coxeter group B_1. This calculation and a similar one for B_0 have been used to prove that B_1' / B_1'' ≅ Z_2^9 × Z^3 and B_0' / B_0'' ≅ Z_2^91 × Z^27 as stated in in [FJNT95, Proposition 5].

    gap> # Define the Coxeter group E1.
    gap> F := FreeGroup( "x1", "x2", "x3", "x4", "x5" );
    <free group on the generators [ x1, x2, x3, x4, x5 ]>
    gap> x1 := F.1;; x2 := F.2;; x3 := F.3;; x4 := F.4;; x5 := F.5;;
    gap> rels := [ x1^2, x2^2, x3^2, x4^2, x5^2,
    >  (x1 * x3)^2, (x2 * x4)^2, (x1 * x2)^3, (x2 * x3)^3, (x3 * x4)^3,
    >  (x4 * x1)^3, (x1 * x5)^3, (x2 * x5)^2, (x3 * x5)^3, (x4 * x5)^2,
    >  (x1 * x2 * x3 * x4 * x3 * x2)^2 ];;
    gap> E1 := F / rels;
    <fp group on the generators [ x1, x2, x3, x4, x5 ]>
    gap> x1 := E1.1;; x2 := E1.2;; x3 := E1.3;; x4 := E1.4;; x5 := E1.5;;
    gap> # Get normal subgroup generators for B1.
    gap> H := Subgroup( E1, [ x5 * x2^-1, x5 * x4^-1 ] );;
    gap> # Compute the abelian invariants of B1/B1'.
    gap> A := AbelianInvariantsNormalClosureFpGroup( E1, H );
    [ 2, 2, 2, 2, 2, 2, 2, 2 ]
    gap> # Compute a presentation for B1.
    gap> P := PresentationNormalClosure( E1, H );
    <presentation with 18 gens and 46 rels of total length 132>
    gap> SimplifyPresentation( P );
    #I  there are 8 generators and 30 relators of total length 148
    gap> B1 := FpGroupPresentation( P );
    <fp group on the generators [ _x1, _x2, _x3, _x4, _x6, _x7, _x8, _x11
     ]>
    gap> # Compute normal subgroup generators for B1'.
    gap> gens := GeneratorsOfGroup( B1 );;
    gap> numgens := Length( gens );;
    gap> comms := [ ];;
    gap> for i in [ 1 .. numgens - 1 ] do
    >     for j in [i+1 .. numgens ] do
    >         Add( comms, Comm( gens[i], gens[j] ) );
    >     od;
    > od;
    gap> # Compute the abelian invariants of B1'/B1".
    gap> K := Subgroup( B1, comms );;
    gap> A := AbelianInvariantsNormalClosureFpGroup( B1, K );
    [ 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]

#### 47.16 Testing Finiteness of Finitely Presented Groups

As a consequence of the algorithmic insolvabilities mentioned in the introduction to this chapter, there cannot be a general method that will test whether a given finitely presented group is actually finite.

Therefore testing the finiteness of a finitely presented group can be problematic. What GAP actually does upon a call of `IsFinite` (30.4-2) (or if it is –probably implicitly– asked for a faithful permutation representation) is to test whether it can find (via coset enumeration) a cyclic subgroup of finite index. If it can, it rewrites the presentation to this subgroup. Since the subgroup is cyclic, its size can be checked easily from the resulting presentation, the size of the whole group is the product of the index and the subgroup size. Since however no bound for the index of such a subgroup (if any exist) is known, such a test might continue unsuccessfully until memory is exhausted.

On the other hand, a couple of methods exist, that might prove that a group is infinite. Again, none is guaranteed to work in every case:

The first method is to find (for example via the low index algorithm, see `LowIndexSubgroupsFpGroup` (47.10-1)) a subgroup U such that [U:U'] is infinite. If U has finite index, this can be checked by `IsInfiniteAbelianizationGroup` (47.16-1).

Note that this test has been done traditionally by checking the `AbelianInvariants` (39.16-1) (see section 47.15) of U, `IsInfiniteAbelianizationGroup` (47.16-1) does a similar calculation but stops as soon as it is known whether 0 is an invariant without computing the actual values. This can be notably faster.

Another method is based on p-group quotients, see `NewmanInfinityCriterion` (47.16-2).

##### 47.16-1 IsInfiniteAbelianizationGroup

`‣ IsInfiniteAbelianizationGroup`( G ) | ( attribute )

returns true if the commutator factor group G/G' is infinite. This might be done without computing the full structure of the commutator factor group.

##### 47.16-2 NewmanInfinityCriterion

`‣ NewmanInfinityCriterion`( G, p ) | ( function )

Let G be a finitely presented group and p a prime that divides the order of the commutator factor group of G. This function applies an infinity criterion due to M. F. Newman [New90] to G. (See [Joh97, chapter 16] for a more explicit description.) It returns `true` if the criterion succeeds in proving that G is infinite and `fail` otherwise.

Note that the criterion uses the number of generators and relations in the presentation of G. Reduction of the presentation via Tietze transformations (`IsomorphismSimplifiedFpGroup` (47.12-1)) therefore might produce an isomorphic group, for which the criterion will work better.

    gap> g:=FibonacciGroup(2,9);
    <fp group on the generators [ f1, f2, f3, f4, f5, f6, f7, f8, f9 ]>
    gap> hom:=EpimorphismNilpotentQuotient(g,2);;
    gap> k:=Kernel(hom);;
    gap> Index(g,k);
    152
    gap> AbelianInvariants(k);
    [ 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5 ]
    gap> NewmanInfinityCriterion(Kernel(hom),5);
    true

This proves that the subgroup `k` (and thus the whole group `g`) is infinite. (This is the original example from [New90].)

[Top of Book] [Contents] [Previous Chapter] [Next Chapter]
Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 Bib Ind

generated by GAPDoc2HTML