GAP (nq) - Chapter 3: The Functions of the Package

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3 The Functions of the Package
3.1 Nilpotent Quotients of Finitely Presented Groups
3.1-1 NilpotentQuotient
3.1-2 NilpotentEngelQuotient
3.1-3 NqEpimorphismNilpotentQuotient
3.1-4 LowerCentralFactors
3.2 Expression Trees
3.2-1 ExpressionTrees
3.2-2 EvaluateExpTree
3.3 Auxiliary Functions
3.3-1 NqReadOutput
3.3-2 NqStringFpGroup
3.3-3 NqStringExpTrees
3.3-4 NqElementaryDivisors
3.4 Global Variables
3.4-1 NqRuntime
3.4-2 NqDefaultOptions
3.4-3 NqGlobalVariables
3.5 Diagnostic Output

### 3 The Functions of the Package

#### 3.1 Nilpotent Quotients of Finitely Presented Groups

##### 3.1-1 NilpotentQuotient

`‣ NilpotentQuotient`( [output-file, ]fp-group[, id-gens][, c] ) | ( function )

`‣ NilpotentQuotient`( [output-file, ]input-file[, c] ) | ( function )

The parameter `fp-group` is either a finitely presented group or a record specifying a presentation by expression trees (see section 2.6). The parameter `input-file` is a string specifying the name of a file containing a finite presentation in the input format (cf. section 2.8) of the ANU NQ. Such a file can be prepared by a text editor or with the help of the function `NqStringFpGroup` (3.3-2).

Let \(G\) be the group defined by `fp-group` or the group defined in `input-file`. The function computes a nilpotent presentation for \(G/\gamma_{c+1}(G)\) if the optional parameter `c` is specified. If `c` is not given, then the function attempts to compute the largest nilpotent quotient of \(G\) and it will terminate only if \(G\) has a largest nilpotent quotient. See section 3.5 for a possibility to follow the progress of the computation.

The optional argument `id-gens` is a list of generators of the free group underlying the finitely presented group `fp-group`. The generators in this list are treated as identical generators. Consequently, all relations of the `fp-group` involving these generators are treated as identical relations for these generators.

In addition to the arguments explained above, the function accepts the following options as shown in the first example below:

- `group` This option can be used instead of the parameter `fp-group`.

- `input\_string` This option can be used to specify a finitely presented group by a string in the input format of the standalone program.

- `input\_file` This option specifies a file with input for the standalone program.

- `output\_file` This option specifies a file for the output of the standalone.

- `idgens` This options specifies a list of identical generators.

- `class` This option specifies the nilpotency class up to which the nilpotent quotient will be computed.

The following example computes the class-5 quotient of the free group on two generators.

    gap> F := FreeGroup( 2 );
    <free group on the generators [ f1, f2 ]>
    gap> ## Equivalent to:  NilpotentQuotient( : group := F, class := 5 );
    gap> ##                 NilpotentQuotient( F : class := 5 );
    gap> H := NilpotentQuotient( F, 5 );
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
    gap> lcs := LowerCentralSeries( H );;
    gap> for i in [1..5] do Print( lcs[i] / lcs[i+1], "\n" ); od;
    Pcp-group with orders [ 0, 0 ]
    Pcp-group with orders [ 0 ]
    Pcp-group with orders [ 0, 0 ]
    Pcp-group with orders [ 0, 0, 0 ]
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0 ]

Note that the lower central series in the example is part of the data returned by the standalone program. Therefore, the execution of the function LowerCentralSeries takes no time.

The next example computes the class-4 quotient of the infinite dihedral group. The group is soluble but not nilpotent. The first factor of its lower central series is a Klein four group and all the other factors are cyclic or order \(2\).

    gap> F := FreeGroup( 2 );
    <free group on the generators [ f1, f2 ]>
    gap> G := F / [F.1^2, F.2^2];
    <fp group on the generators [ f1, f2 ]>
    gap> H := NilpotentQuotient( G, 4 );
    Pcp-group with orders [ 2, 2, 2, 2, 2 ]
    gap> lcs := LowerCentralSeries( H );;
    gap> for i in [1..Length(lcs)-1] do
    >       Print( AbelianInvariants(lcs[i] / lcs[i+1]), "\n" );
    > od;
    [ 2, 2 ]
    [ 2 ]
    [ 2 ]
    [ 2 ]
    gap>

In the following example identical generators are used in order to express the fact that the group is nilpotent of class \(3\). A group is nilpotent of class \(3\) if it satisfies the identical relation \([x_1,x_2,x_3,x_4]=1\) (cf. Section 2.5). The result is the free nilpotent group of class \(3\) on two generators.

    gap> F := FreeGroup( "a", "b", "w", "x", "y", "z" );
    <free group on the generators [ a, b, w, x, y, z ]>
    gap> G := F / [ LeftNormedComm( [F.3,F.4,F.5,F.6] ) ];
    <fp group of size infinity on the generators [ a, b, w, x, y, z ]>
    gap> ## The following is equivalent to:
    gap> ##   NilpotentQuotient( G : idgens := [F.3,F.4,F.5,F.6] );
    gap> H := NilpotentQuotient( G, [F.3,F.4,F.5,F.6] );
    Pcp-group with orders [ 0, 0, 0, 0, 0 ]
    gap> NilpotencyClassOfGroup(H);
    3
    gap> LowerCentralSeries(H);
    [ Pcp-group with orders [ 0, 0, 0, 0, 0 ], Pcp-group with orders [ 0, 0, 0 ],
      Pcp-group with orders [ 0, 0 ], Pcp-group with orders [  ] ]

The following example uses expression trees in order to specify the third Engel law for the free group on \(3\) generators.

    gap> et := ExpressionTrees( 5 );
    [ x1, x2, x3, x4, x5 ]
    gap> comm := LeftNormedComm( [et[1], et[2], et[2], et[2]] );
    Comm( x1, x2, x2, x2 )
    gap> G := rec( generators := et, relations := [comm] );
    rec( generators := [ x1, x2, x3, x4, x5 ],
      relations := [ Comm( x1, x2, x2, x2 ) ] )
    gap> H := NilpotentQuotient( G : idgens := [et[1],et[2]] );
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 2,
      0, 6, 6, 0, 0, 2, 10, 10, 10 ]
    gap> TorsionSubgroup( H );
    Pcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 10, 10, 10 ]
    gap> lcs := LowerCentralSeries( H );;
    gap> NilpotencyClassOfGroup( H );
    5
    gap> for i in [1..5] do Print( lcs[i] / lcs[i+1], "\n" ); od;
    Pcp-group with orders [ 0, 0, 0 ]
    Pcp-group with orders [ 0, 0, 0 ]
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0 ]
    Pcp-group with orders [ 2, 4, 2, 2, 0, 6, 6, 0, 0, 2 ]
    Pcp-group with orders [ 10, 10, 10 ]
    gap> for i in [1..5] do Print( AbelianInvariants(lcs[i]/lcs[i+1]), "\n" ); od;
    [ 0, 0, 0 ]
    [ 0, 0, 0 ]
    [ 0, 0, 0, 0, 0, 0, 0, 0 ]
    [ 2, 2, 2, 2, 2, 2, 2, 0, 0, 0 ]
    [ 10, 10, 10 ]

The example above also shows that the relative orders of an abelian polycyclic group need not be the abelian invariants (elementary divisors) of the group. Each zero corresponds to a generator of infinite order. The number of zeroes is always correct.

##### 3.1-2 NilpotentEngelQuotient

`‣ NilpotentEngelQuotient`( [output-file, ]fp-group, n[, id-gens][, c] ) | ( function )

`‣ NilpotentEngelQuotient`( [output-file, ]input-file, n[, c] ) | ( function )

This function is a special version of `NilpotentQuotient` (3.1-1) which enforces the \(n\)-th Engel identity on the nilpotent quotients of the group specified by `fp-group` or by `input-file`. It accepts the same options as `NilpotentQuotient`.

The Engel condition can also be enforced by using identical generators and the Engel law and `NilpotentQuotient` (3.1-1). See the examples there.

The following example computes the relatively free fifth Engel group on two generators, determines its (normal) torsion subgroup and computes the corresponding quotient group. The quotient modulo the torsion subgroup is torsion-free. Therefore, there is a nilpotent presentation without power relations. The example computes a nilpotent presentation for the torsion free factor group through the upper central series. The factors of the upper central series in a torsion free group are torsion free. In this way one obtains a set of generators of infinite order and the resulting nilpotent presentation has no power relations.

    gap> G := NilpotentEngelQuotient( FreeGroup(2), 5 );
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 10,
      0, 0, 30, 0, 3, 3, 10, 2, 0, 6, 0, 0, 30, 2, 0, 9, 3, 5, 2, 6, 2, 10, 5, 5,
      2, 0, 3, 3, 3, 3, 3, 5, 5, 3, 3 ]
    gap> NilpotencyClassOfGroup(G);
    9
    gap> T := TorsionSubgroup( G );
    Pcp-group with orders [ 3, 3, 2, 2, 3, 3, 2, 9, 3, 5, 2, 3, 2, 10, 5, 2, 3,
      3, 3, 3, 3, 5, 5, 3, 3 ]
    gap> IsAbelian( T );
    true
    gap> AbelianInvariants( T );
    [ 3, 3, 3, 3, 3, 3, 3, 3, 30, 30, 30, 180, 180 ]
    gap> H := G / T;
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 10,
      0, 0, 30, 0, 5, 0, 2, 0, 0, 10, 0, 2, 5, 0 ]
    gap> H := PcpGroupBySeries( UpperCentralSeries(H), "snf" );
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0 ]
    gap> ucs := UpperCentralSeries( H );;
    gap> for i in [1..NilpotencyClassOfGroup(H)] do
    > 	Print( ucs[i]/ucs[i+1], "\n" );
    > od;
    Pcp-group with orders [ 0, 0 ]
    Pcp-group with orders [ 0 ]
    Pcp-group with orders [ 0, 0 ]
    Pcp-group with orders [ 0, 0, 0 ]
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0 ]
    Pcp-group with orders [ 0, 0, 0, 0 ]
    Pcp-group with orders [ 0, 0 ]
    Pcp-group with orders [ 0, 0, 0 ]

##### 3.1-3 NqEpimorphismNilpotentQuotient

`‣ NqEpimorphismNilpotentQuotient`( [output-file, ]fp-group[, id-gens][, c] ) | ( function )

This function computes an epimorphism from the group \(G\) given by the finite presentation `fp-group` onto \(G/\gamma_{c+1}(G).\) If `c` is not given, then the largest nilpotent quotient of \(G\) is computed and an epimorphism from \(G\) onto the largest nilpotent quotient of \(G\). If \(G\) does not have a largest nilpotent quotient, the function will not terminate if \(c\) is not given.

The optional argument `id-gens` is a list of generators of the free group underlying the finitely presented group `fp-group`. The generators in this list are treated as identical generators. Consequently, all relations of the `fp-group` involving these generators are treated as identical relations for these generators.

If identical generators are specified, then the epimorphism returned maps the group generated by the `non-identical' generators onto the nilpotent factor group. See the last example below.

The function understands the same options as the function `NilpotentQuotient` (3.1-1).

    gap> F := FreeGroup(3);
    <free group on the generators [ f1, f2, f3 ]>
    gap> phi := NqEpimorphismNilpotentQuotient( F, 5 );
    [ f1, f2, f3 ] -> [ g1, g2, g3 ]
    gap> Image( phi, LeftNormedComm( [F.3, F.2, F.1] ) );
    g12
    gap> F := FreeGroup( "a", "b" );
    <free group on the generators [ a, b ]>
    gap> G := F / [ F.1^2, F.2^2 ];
    <fp group on the generators [ a, b ]>
    gap> phi := NqEpimorphismNilpotentQuotient( G, 4 );
    [ a, b ] -> [ g1, g2 ]
    gap> Image( phi, Comm(G.1,G.2) );
    g3*g4
    gap> F := FreeGroup( "a", "b", "u", "v", "x" );
    <free group on the generators [ a, b, u, v, x ]>
    gap> a := F.1;; b := F.2;; u := F.3;; v := F.4;; x := F.5;;
    gap> G := F / [ x^5, LeftNormedComm( [u,v,v,v] ) ];
    <fp group of size infinity on the generators [ a, b, u, v, x ]>
    gap> phi := NqEpimorphismNilpotentQuotient( G : idgens:=[u,v,x], class:=5 );
    [ a, b ] -> [ g1, g2 ]
    gap> U := Source(phi);
    Group([ a, b ])
    gap> ImageElm( phi, LeftNormedComm( [U.1*U.2, U.2^-1,U.2^-1,U.2^-1,] ) );
    id

Note that the last epimorphism is a map from the group generated by \(a\) and \(b\) onto the nilpotent quotient. The identical generators are used only to formulate the identical relator. They are not generators of the group \(G\). Also note that the left-normed commutator above is mapped to the identity as \(G\) satisfies the specified identical law.

##### 3.1-4 LowerCentralFactors

`‣ LowerCentralFactors`( ... ) | ( function )

This function accepts the same arguments and options as `NilpotentQuotient` (3.1-1) and returns a list containing the abelian invariants of the central factors in the lower central series of the specified group.

    gap> LowerCentralFactors( FreeGroup(2), 6 );
    [ [ 0, 0 ], [ 0 ], [ 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ]

#### 3.2 Expression Trees

##### 3.2-1 ExpressionTrees

`‣ ExpressionTrees`( m[, prefix] ) | ( function )

`‣ ExpressionTrees`( str1, str2, str3, ... ) | ( function )

The argument `m` must be a positive integer. The function returns a list with `m` expression tree symbols named x1, x2,... The optional parameter `prefix` must be a string and is used instead of `x` if present.

Alternatively, the function can be executed with a list of strings `str1`, `str2`, .... It returns a list of symbols with these strings as names.

The following operations are defined for expression trees: multiplication, inversion, exponentiation, forming commutators, forming conjugates.

    gap> t := ExpressionTrees( 3 );
    [ x1, x2, x3 ]
    gap> tree := Comm( t[1], t[2] )^3/LeftNormedComm( [t[1],t[2],t[3],t[1]] );
    Comm( x1, x2 )^3/Comm( x1, x2, x3, x1 )
    gap> t := ExpressionTrees( "a", "b", "x" );
    [ a, b, x ]
    gap> tree := Comm( t[1], t[2] )^3/LeftNormedComm( [t[1],t[2],t[3],t[1]] );
    Comm( a, b )^3/Comm( a, b, x, a )

##### 3.2-2 EvaluateExpTree

`‣ EvaluateExpTree`( tree, symbols, values ) | ( function )

The argument `tree` is an expression tree followed by the list of those symbols `symbols` from which the expression tree is built up. The argument `values` is a list containing a constant for each symbol. The function substitutes each value for the corresponding symbol and computes the resulting value for `tree`.

    gap> F := FreeGroup( 3 );
    <free group on the generators [ f1, f2, f3 ]>
    gap> t := ExpressionTrees( "a", "b", "x" );
    [ a, b, x ]
    gap> tree := Comm( t[1], t[2] )^3/LeftNormedComm( [t[1],t[2],t[3],t[1]] );
    Comm( a, b )^3/Comm( a, b, x, a )
    gap> EvaluateExpTree( tree, t, GeneratorsOfGroup(F) );
    f1^-1*f2^-1*f1*f2*f1^-1*f2^-1*f1*f2*f1^-1*f2^-1*f1*f2*f1^-1*f3^-1*f2^-1*f1^
    -1*f2*f1*f3*f1^-1*f2^-1*f1*f2*f1*f2^-1*f1^-1*f2*f1*f3^-1*f1^-1*f2^-1*f1*f2*f3

#### 3.3 Auxiliary Functions

##### 3.3-1 NqReadOutput

`‣ NqReadOutput`( stream ) | ( function )

The only argument `stream` is an output stream of the ANU NQ. The function reads the stream and returns a record that has a component for each global variable used in the output of the ANU NQ, see `NqGlobalVariables` (3.4-3).

##### 3.3-2 NqStringFpGroup

`‣ NqStringFpGroup`( fp-group[, idgens] ) | ( function )

The function takes a finitely presented group `fp-group` and returns a string in the input format of the ANU NQ. If the list `idgens` is present, then it must contain generators of the free group underlying the finitely presented group `FreeGroupOfFpGroup` (Reference: FreeGroupOfFpGroup). The generators in `idgens` are treated as identical generators.

    gap> F := FreeGroup(2);
    <free group on the generators [ f1, f2 ]>
    gap> G := F / [F.1^2, F.2^2, (F.1*F.2)^4];
    <fp group on the generators [ f1, f2 ]>
    gap> NqStringFpGroup( G );
    "< x1, x2 |\n    x1^2,\n    x2^2,\n    x1*x2*x1*x2*x1*x2*x1*x2\n>\n"
    gap> Print( last );
    < x1, x2 |
        x1^2,
        x2^2,
        x1*x2*x1*x2*x1*x2*x1*x2
    >
    gap> PrintTo( "dihedral", last );
    gap> ## The following is equivalent to:
    gap> ##     NilpotentQuotient( : input_file := "dihedral" );
    gap> NilpotentQuotient( "dihedral" );
    Pcp-group with orders [ 2, 2, 2 ]
    gap> Exec( "rm dihedral" );
    gap> F := FreeGroup(3);
    <free group on the generators [ f1, f2, f3 ]>
    gap> H := F / [ LeftNormedComm( [F.2,F.1,F.1] ),
    >               LeftNormedComm( [F.2,F.1,F.2] ), F.3^7 ];
    <fp group on the generators [ f1, f2, f3 ]>
    gap> str := NqStringFpGroup( H, [F.3] );
    "< x1, x2; x3 |\n    x1^-1*x2^-1*x1*x2*x1^-1*x2^-1*x1^-1*x2*x1^2,\n    x1^-1*x\
    2^-1*x1*x2^-1*x1^-1*x2*x1*x2,\n    x3^7\n>\n"
    gap> NilpotentQuotient( : input_string := str );
    Pcp-group with orders [ 7, 7, 7 ]

##### 3.3-3 NqStringExpTrees

`‣ NqStringExpTrees`( fp-group[, idgens] ) | ( function )

The function takes a finitely presented group `fp-group` given in terms of expression trees and returns a string in the input format of the ANU NQ. If the list `idgens` is present, then it must contain a sublist of the generators of the presentation. The generators in `idgens` are treated as identical generators.

    gap> x := ExpressionTrees( 2 );
    [ x1, x2 ]
    gap> rels := [x[1]^2, x[2]^2, (x[1]*x[2])^5];
    [ x1^2, x2^2, (x1*x2)^5 ]
    gap> NqStringExpTrees( rec( generators := x, relations := rels ) );
    "< x1, x2 |\n    x1^2,\n    x2^2,\n    (x1*x2)^5\n>\n"
    gap> Print( last );
    < x1, x2 |
        x1^2,
        x2^2,
        (x1*x2)^5
    >
    gap> x := ExpressionTrees( 3 );
    [ x1, x2, x3 ]
    gap> rels := [LeftNormedComm( [x[2],x[1],x[1]] ),
    >             LeftNormedComm( [x[2],x[1],x[2]] ), x[3]^7 ];
    [ Comm( x2, x1, x1 ), Comm( x2, x1, x2 ), x3^7 ]
    gap> NqStringExpTrees( rec( generators := x, relations := rels ) );
    "< x1, x2, x3 |\n    [ x2, x1, x1 ],\n    [ x2, x1, x2 ],\n    x3^7\n>\n"
    gap> Print( last );
    < x1, x2, x3 |
        [ x2, x1, x1 ],
        [ x2, x1, x2 ],
        x3^7
    >

##### 3.3-4 NqElementaryDivisors

`‣ NqElementaryDivisors`( int-mat ) | ( function )

The function `ElementaryDivisorsMat` (Reference: ElementaryDivisorsMat) only returns the non-zero elementary divisors of an integer matrix. This function computes the elementary divisors of `int-mat` and adds the appropriate number of zeroes in order to make it easier to recognize the isomorphism type of the abelian group presented by the integer matrix. At the same time ones are stripped from the list of elementary divisors.

#### 3.4 Global Variables

##### 3.4-1 NqRuntime

`‣ NqRuntime` | ( global variable )

This variable contains the number of milliseconds of runtime of the last call of ANU NQ.

    gap> NilpotentEngelQuotient( FreeGroup(2), 5 );
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 10,
      0, 0, 30, 0, 3, 3, 10, 2, 0, 6, 0, 0, 30, 2, 0, 9, 3, 5, 2, 6, 2, 10, 5, 5,
      2, 0, 3, 3, 3, 3, 3, 5, 5, 3, 3 ]
    gap> NqRuntime;
    18200

##### 3.4-2 NqDefaultOptions

`‣ NqDefaultOptions` | ( global variable )

This variable contains a list of strings which are the standard command line options passed to the ANU NQ in each call. Modifying this variable can be used to pass additional options to the ANU NQ.

    gap> NqDefaultOptions;
    [ "-g", "-p", "-C", "-s" ]

The option -g causes the ANU NQ to produce output in GAP-format. The option -p prevents the ANU NQ from listing the pc-presentation of the nilpotent quotient at the end of the calculation. The option -C invokes the combinatorial collector. The option -s is effective only in conjunction with options for computing with Engel identities and instructs the ANU NQ to use only semigroup words in the generators as instances of an Engel law.

##### 3.4-3 NqGlobalVariables

`‣ NqGlobalVariables` | ( global variable )

This variable contains a list of strings with the names of the global variables that are used in the output stream of the ANU NQ. While the output stream is read, these global variables are assigned new values. To avoid overwriting these variables in case they contain values, their contents is saved before reading the output stream and restored afterwards.

#### 3.5 Diagnostic Output

While the standalone program is running it can be asked to display progress information. This is done by setting the info class `InfoNQ` to \(1\) via the function `SetInfoLevel` (Reference: InfoLevel).

    gap> NilpotentQuotient(FreeGroup(2),5);
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
    gap> SetInfoLevel( InfoNQ, 1 );
    gap> NilpotentQuotient(FreeGroup(2),5);
    #I  Class 1: 2 generators with relative orders  0 0
    #I  Class 2: 1 generators with relative orders: 0
    #I  Class 3: 2 generators with relative orders: 0 0
    #I  Class 4: 3 generators with relative orders: 0 0 0
    #I  Class 5: 6 generators with relative orders: 0 0 0 0 0 0
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
    gap> SetInfoLevel( InfoNQ, 0 );

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