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### 5 Interactive ANUPQ functions

Here we describe the interactive functions defined by the ANUPQ package, i.e. the functions that manipulate and initiate interactive ANUPQ processes. These are functions that extract information via a dialogue with a running pq process (process used in the UNIX sense). Occasionally, a user needs the "next step"; the functions provided in this chapter make use of data from previous steps retained by the pq program, thus allowing the user to interact with the pq program like one can when one uses the pq program as a stand-alone (see guide.dvi in the standalone-doc directory).

An interactive ANUPQ process is initiated by PqStart and terminated via PqQuit; these functions are described in ection Starting and Stopping Interactive ANUPQ Processes.

Each interactive ANUPQ function that manipulates an already started interactive ANUPQ process, has a form where the first argument is the integer i returned by the initiating PqStart command, and a second form with one argument fewer (where the integer i is discovered by a default mechanism, namely by determining the least integer i for which there is a currently active interactive ANUPQ process). We will thus commonly say that "for the ith (or default) interactive ANUPQ process" a certain function performs a given action. In each case, it is an error if i is not the index of an active interactive process, or there are no current active interactive processes.

Notes: The global method of passing options (via PushOptions), should not be used with any of the interactive functions. In fact, the OptionsStack should be empty at the time any of the interactive functions is called.

On quitting GAP, PqQuitAll(); is executed, which terminates all active interactive ANUPQ processes. If GAP is killed without quitting, before all interactive ANUPQ processes are terminated, zombie processes (still living child processes whose parents have died), may result. Since zombie processes do consume resources, in such an event, the responsible computer user should seek out and terminate those zombie processes (e.g. on Linux: ps xw | grep pq gives you information on the pq processes corresponding to any interactive ANUPQ processes started in a GAP session; you can then do kill N for each number N appearing in the first column of this output).

#### 5.1 Starting and Stopping Interactive ANUPQ Processes

##### 5.1-1 PqStart
 ‣ PqStart( G, workspace: options ) ( function )
 ‣ PqStart( G: options ) ( function )
 ‣ PqStart( workspace: options ) ( function )
 ‣ PqStart( : options ) ( function )

activate an iostream for an interactive ANUPQ process (i.e. PqStart starts up a pq process and opens a GAP iostream to "talk" to that process) and returns an integer i that can be used to identify that process. The argument G should be an fp group or pc group that the user intends to manipulate using interactive ANUPQ functions. If the function is called without specifying G, a group can be read in by using the function PqRestorePcPresentation (see PqRestorePcPresentation (5.6-3)). If PqStart is given an integer argument workspace, then the pq program is started up with a workspace (an integer array) of size workspace (i.e. 4 × workspace bytes in a 32-bit environment); otherwise, the pq program sets a default workspace of 10000000.

The only options currently recognised by PqStart are Prime, Exponent and Relators (see Chapter ANUPQ Options for detailed descriptions of these options) and if provided they are essentially global for the interactive ANUPQ process, except that any interactive function interacting with the process and passing new values for these options will over-ride the global values.

##### 5.1-2 PqQuit
 ‣ PqQuit( i ) ( function )
 ‣ PqQuit( ) ( function )

closes the stream of the ith or default interactive ANUPQ process and unbinds its ANUPQData.io record.

Note: It can happen that the pq process, and hence the GAP iostream assigned to communicate with it, can die, e.g. by the user typing a Ctrl-C while the pq process is engaged in a long calculation. IsPqProcessAlive (see IsPqProcessAlive (5.2-3)) is provided to check the status of the GAP iostream (and hence the status of the pq process it was communicating with).

##### 5.1-3 PqQuitAll
 ‣ PqQuitAll( ) ( function )

is provided as a convenience, to terminate all active interactive ANUPQ processes with a single command. It is equivalent to executing PqQuit(i) for all active interactive ANUPQ processes i (see PqQuit (5.1-2)).

#### 5.2 Interactive ANUPQ Process Utility Functions

##### 5.2-1 PqProcessIndex
 ‣ PqProcessIndex( i ) ( function )
 ‣ PqProcessIndex( ) ( function )

With argument i, which must be a positive integer, PqProcessIndex returns i if it corresponds to an active interactive process, or raises an error. With no arguments it returns the default active interactive process or returns fail and emits a warning message to Info at InfoANUPQ or InfoWarning level 1.

Note: Essentially, an interactive ANUPQ process i is "active" if ANUPQData.io[i] is bound (i.e. we still have some data telling us about it). Also see PqStart (5.1-1).

##### 5.2-2 PqProcessIndices
 ‣ PqProcessIndices( ) ( function )

returns the list of integer indices of all active interactive ANUPQ processes (see PqProcessIndex (5.2-1) for the meaning of "active").

##### 5.2-3 IsPqProcessAlive
 ‣ IsPqProcessAlive( i ) ( function )
 ‣ IsPqProcessAlive( ) ( function )

return true if the GAP iostream of the ith (or default) interactive ANUPQ process started by PqStart is alive (i.e. can still be written to), or false, otherwise. (See the notes for PqStart (5.1-1) and PqQuit (5.1-2).)

If the user does not yet have a gap> prompt then usually the pq program is still away doing something and an ANUPQ interface function is still waiting for a reply. Typing a Ctrl-C (i.e. holding down the Ctrl key and typing c) will stop the waiting and send GAP into a break-loop, from which one has no option but to quit;. The typing of Ctrl-C, in such a circumstance, usually causes the stream of the interactive ANUPQ process to die; to check this we provide IsPqProcessAlive (see IsPqProcessAlive).

The GAP iostream of an interactive ANUPQ process will also die if the pq program has a segmentation fault. We do hope that this never happens to you, but if it does and the failure is reproducible, then it's a bug and we'd like to know about it. Please read the README that comes with the ANUPQ package to find out what to include in a bug report and who to email it to.

#### 5.3 Interactive Versions of Non-interactive ANUPQ Functions

##### 5.3-1 Pq
 ‣ Pq( i: options ) ( function )
 ‣ Pq( : options ) ( function )

return, for the fp or pc group (let us call it F), of the ith or default interactive ANUPQ process, the p-quotient of F specified by options, as a pc group; F must previously have been given (as first argument) to PqStart to start the interactive ANUPQ process (see PqStart (5.1-1)) or restored from file using the function PqRestorePcPresentation (see PqRestorePcPresentation (5.6-3)). Following the colon options is a selection of the options listed for the non-interactive Pq function (see Pq (4.1-1)), separated by commas like record components (see Section Reference: Function Call With Options in the GAP Reference Manual), except that the options SetupFile or PqWorkspace are ignored by the interactive Pq, and RedoPcp is an option only recognised by the interactive Pq i.e. the following options are recognised by the interactive Pq function:

• Prime := p

• ClassBound := n

• Exponent := n

• Relators := rels

• Metabelian

• Identities := funcs

• GroupName := name

• OutputLevel := n

• RedoPcp

Detailed descriptions of the above options may be found in Chapter ANUPQ Options.

As a minimum the Pq function must have a value for the Prime option, though Prime need not be passed again in the case it has previously been provided, e.g. to PqStart (see PqStart (5.1-1)) when starting the interactive process.

The behaviour of the interactive Pq function depends on the current state of the pc presentation stored by the pq program:

1. If no pc presentation has yet been computed (the case immediately after the PqStart call initiating the process) then the quotient group of the input group of the process of largest lower exponent-p class bounded by the value of the ClassBound option (see 6.2) is returned.

2. If the current pc presentation of the process was determined by a previous call to Pq or PqEpimorphism, and the current call has a larger value ClassBound then the class is extended as much as is possible and the quotient group of the input group of the process of the new lower exponent-p class is returned.

3. If the current pc presentation of the process was determined by a previous call to PqPCover then a consistent pc presentation of a quotient for the current class is determined before proceeding as in 2.

4. If the RedoPcp option is supplied the current pc presentation is scrapped, all options must be re-supplied (in particular, Prime must be supplied) and then the Pq function proceeds as in 1.

See Section Attributes and a Property for fp and pc p-groups for the attributes and property NuclearRank, MultiplicatorRank and IsCapable which may be applied to the group returned by Pq.

The following is one of the examples for the non-interactive Pq redone with the interactive version. Also, we set the option OutputLevel to 1 (see 6.2), in order to see the orders of the quotients of all the classes determined, and we set the InfoANUPQ level to 2 (see InfoANUPQ (3.3-1)), so that we catch the timing information.

gap> F := FreeGroup("a", "b");; a := F.1;; b := F.2;;
gap> G := F / [a^4, b^4];
<fp group on the generators [ a, b ]>
gap> PqStart(G);
1
gap> SetInfoLevel(InfoANUPQ, 2); #To see timing information
gap> Pq(: Prime := 2, ClassBound := 3, OutputLevel := 1 );
#I  Lower exponent-2 central series for [grp]
#I  Group: [grp] to lower exponent-2 central class 1 has order 2^2
#I  Group: [grp] to lower exponent-2 central class 2 has order 2^5
#I  Group: [grp] to lower exponent-2 central class 3 has order 2^8
#I  Computation of presentation took 0.00 seconds
<pc group of size 256 with 8 generators>


##### 5.3-2 PqEpimorphism
 ‣ PqEpimorphism( i: options ) ( function )
 ‣ PqEpimorphism( : options ) ( function )

return, for the fp or pc group (let us call it F), of the ith or default interactive ANUPQ process, an epimorphism from F onto the p-quotient of F specified by options; F must previously have been given (as first argument) to PqStart to start the interactive ANUPQ process (see PqStart (5.1-1)). Since the underlying interactions with the pq program effected by the interactive PqEpimorphism are identical to those effected by the interactive Pq, everything said regarding the requirements and behaviour of the interactive Pq function (see Pq (5.3-1)) is also the case for the interactive PqEpimorphism.

Note: See Section Attributes and a Property for fp and pc p-groups for the attributes and property NuclearRank, MultiplicatorRank and IsCapable which may be applied to the image group of the epimorphism returned by PqEpimorphism.

##### 5.3-3 PqPCover
 ‣ PqPCover( i: options ) ( function )
 ‣ PqPCover( : options ) ( function )

return, for the fp or pc group of the ith or default interactive ANUPQ process, the p-covering group of the p-quotient Pq(i : options) or Pq(: options), modulo the following:

1. If no pc presentation has yet been computed (the case immediately after the PqStart call initiating the process) and the group F of the process is already a p-group, in the sense that HasIsPGroup(F) and IsPGroup(F) is true, then

Prime

defaults to PrimePGroup(F), if not supplied and HasPrimePGroup(F) = true; and

ClassBound

defaults to PClassPGroup(F) if HasPClassPGroup(F) = true if not supplied, or to the usual default of 63, otherwise.

2. If a pc presentation has been computed and none of options is RedoPcp or if no pc presentation has yet been computed but 1. does not apply then PqPCover(i : options); is equivalent to:

Pq(i : options);
PqPCover(i);

3. If the RedoPcp option is supplied the current pc presentation is scrapped, and PqPCover proceeds as in 1. or 2. but without the RedoPcp option.

##### 5.3-4 PqStandardPresentation
 ‣ PqStandardPresentation( [i]: options ) ( function )
 ‣ StandardPresentation( [i]: options ) ( function )

return, for the ith or default interactive ANUPQ process, the p-quotient of the group F of the process, specified by options, as an fp group which has a standard presentation. Here options is a selection of the options from the following list (see Chapter ANUPQ Options for detailed descriptions); this list is the same as for the non-interactive version of PqStandardPresentation except for the omission of options SetupFile and PqWorkspace (see PqStandardPresentation (4.2-1)).

• Prime := p

• pQuotient := Q

• ClassBound := n

• Exponent := n

• Metabelian

• GroupName := name

• OutputLevel := n

• StandardPresentationFile := filename

Unless F is a pc p-group, or the option Prime has been passed to a previous interactive function for the process to compute a p-quotient for F, the user must supply either the option Prime or the option pQuotient (if both Prime and pQuotient are supplied, the prime p is determined by applying PrimePGroup (see PrimePGroup (Reference: PrimePGroup) in the Reference Manual) to the value of pQuotient).

Taking one of the examples for the non-interactive version of StandardPresentation (see StandardPresentation (4.2-1)) that required two separate calls to the pq program, we now show how it can be done by setting up a dialogue with just the one pq process, using the interactive version of StandardPresentation:

gap> F4 := FreeGroup( "a", "b", "c", "d" );;
gap> a := F4.1;; b := F4.2;; c := F4.3;; d := F4.4;;
gap> G4 := F4 / [ b^4, b^2 / Comm(Comm (b, a), a), d^16,
>                 a^16 / (c * d), b^8 / (d * c^4) ];
<fp group on the generators [ a, b, c, d ]>
gap> SetInfoLevel(InfoANUPQ, 1); #Only essential Info please
gap> PqStart(G4); #Start a new interactive process for a new group
2
gap> K := Pq( 2 : Prime := 2, ClassBound := 1 ); #pq' process no. is 2
<pc group of size 4 with 2 generators>
gap> StandardPresentation( 2 : pQuotient := K, ClassBound := 14 );
<fp group with 53 generators>


Notes

In contrast to the function Pq (see Pq (4.1-1)) which returns a pc group, PqStandardPresentation or StandardPresentation returns an fp group. This is because the output is mainly used for isomorphism testing for which an fp group is enough. However, the presentation is a polycyclic presentation and if you need to do any further computation with this group (e.g. to find the order) you can use the function PcGroupFpGroup (see PcGroupFpGroup (Reference: PcGroupFpGroup) in the GAP Reference Manual) to form a pc group.

If the user does not supply a p-quotient Q via the pQuotient option, and the prime p is either supplied, stored, or F is a pc p-group, then a p-quotient Q is computed. (The value of the prime p is stored if passed initially to PqStart or to a subsequent interactive process.) Note that a stored value for pQuotient (from a prior call to Pq) does not have precedence over a value for the prime p. If the user does supply a p-quotient Q via the pQuotient option, the package AutPGrp is called to compute the automorphism group of Q; an error will occur that asks the user to install the package AutPGrp if the automorphism group cannot be computed.

If any of the interactive functions PqStandardPresentation, StandardPresentation, EpimorphismPqStandardPresentation or EpimorphismStandardPresentation has been called previously for an interactive process, a subsequent call to any of these functions for the same process returns the previously computed value. Note that all these functions compute both an epimorphism and an fp group and store the results in the SPepi and SP fields of the data record associated with the process. See the example for the interactive EpimorphismStandardPresentation (EpimorphismStandardPresentation (5.3-5)).

The attributes and property NuclearRank, MultiplicatorRank and IsCapable are set for the group returned by PqStandardPresentation or StandardPresentation (see Section Attributes and a Property for fp and pc p-groups).

##### 5.3-5 EpimorphismPqStandardPresentation
 ‣ EpimorphismPqStandardPresentation( [i]: options ) ( function )
 ‣ EpimorphismStandardPresentation( [i]: options ) ( method )

Each of the above functions accepts the same arguments and options as the interactive form of StandardPresentation (see StandardPresentation (5.3-4)) and returns an epimorphism from the fp or pc group F of the ith or default interactive ANUPQ process onto the finitely presented group given by a standard presentation, i.e. if S is the standard presentation computed for the p-quotient of F by StandardPresentation then EpimorphismStandardPresentation returns the epimorphism from F to the group with presentation S. The group F must have been given (as first argument) to PqStart to start the interactive ANUPQ process (see PqStart (5.1-1)).

Taking our earlier non-interactive example (see EpimorphismPqStandardPresentation (4.2-2)) and modifying it a little, we illustrate, as for the interactive StandardPresentation (see StandardPresentation (5.3-4)), how something that required two separate calls to the pq program can now be achieved with a dialogue with just one pq process. Also, observe that calls to one of the standard presentation functions (as mentioned in the notes of StandardPresentation (5.3-4)) computes and stores both an fp group with a standard presentation and an epimorphism; subsequent calls to a standard presentation function for the same process simply return the appropriate stored value.

gap> F := FreeGroup(6, "F");;
gap> x := F.1;; y := F.2;; z := F.3;; w := F.4;; a := F.5;; b := F.6;;
gap> R := [x^3 / w, y^3 / w * a^2 * b^2, w^3 / b,
>          Comm (y, x) / z, Comm (z, x), Comm (z, y) / a, z^3 ];
[ F1^3*F4^-1, F2^3*F4^-1*F5^2*F6^2, F4^3*F6^-1, F2^-1*F1^-1*F2*F1*F3^-1,
F3^-1*F1^-1*F3*F1, F3^-1*F2^-1*F3*F2*F5^-1, F3^3 ]
gap> Q := F / R;
<fp group on the generators [ F1, F2, F3, F4, F5, F6 ]>
gap> PqStart( Q );
3
gap> G := Pq( 3 : Prime := 3, ClassBound := 3 );
<pc group of size 729 with 6 generators>
gap> lev := InfoLevel(InfoANUPQ);; # Save current InfoANUPQ level
gap> SetInfoLevel(InfoANUPQ, 2); # To see computation times
gap> # It is not necessary to pass the Prime' option to
gap> # EpimorphismStandardPresentation' since it was previously
gap> # passed to Pq':
gap> phi := EpimorphismStandardPresentation( 3 : ClassBound := 3 );
#I  Class 1 3-quotient and its 3-covering group computed in 0.00 seconds
#I  Order of GL subgroup is 48
#I  No. of soluble autos is 0
#I    dim U = 1  dim N = 3  dim M = 3
#I    nice stabilizer with perm rep
#I  Computing standard presentation for class 2 took 0.00 seconds
#I  Computing standard presentation for class 3 took 0.01 seconds
[ F1, F2, F3, F4, F5, F6 ] -> [ f1*f2^2*f3*f4^2*f5^2, f1*f2*f3*f5, f3^2,
f4*f6^2, f5, f6 ]
gap> # Image of phi should be isomorphic to G ...
gap> # let's check the order is correct:
gap> Size( Image(phi) );
729
gap> # StandardPresentation' and EpimorphismStandardPresentation'
gap> # behave like attributes, so no computation is done when
gap> # either is called again for the same process ...
gap> StandardPresentation( 3 : ClassBound := 3 );
<fp group of size 729 on the generators [ f1, f2, f3, f4, f5, f6 ]>
gap> # No timing data was Info-ed since no computation was done
gap> SetInfoLevel(InfoANUPQ, lev); # Restore previous InfoANUPQ level


A very similar (essential details are the same) example to the above may be executed live, by typing: PqExample( "EpimorphismStandardPresentation-i" );.

Note: The notes for PqStandardPresentation or StandardPresentation (see PqStandardPresentation (5.3-4)) apply also to EpimorphismPqStandardPresentation or EpimorphismStandardPresentation except that their return value is an epimorphism onto an fp group, i.e. one should interpret the phrase "returns an fp group" as "returns an epimorphism onto an fp group" etc.

##### 5.3-6 PqDescendants
 ‣ PqDescendants( i: options ) ( function )
 ‣ PqDescendants( : options ) ( function )

return for the pc group G of the ith or default interactive ANUPQ process, which must be of prime power order with a confluent pc presentation (see IsConfluent (Reference: IsConfluent for pc groups) in the GAP Reference Manual), a list of descendants (pc groups) of G. The group G is usually given as first argument to PqStart when starting the interactive ANUPQ process (see PqStart (5.1-1)). Alternatively, one may initiate the process with an fp group, use Pq interactively (see Pq (5.3-1)) to create a pc group and use PqSetPQuotientToGroup (see PqSetPQuotientToGroup (5.3-7)), which involves no computation, to set the pc group returned by Pq as the group of the process. Note that repeating a call to PqDescendants for the same interactive ANUPQ process simply returns the list of descendants originally calculated; a warning is emitted at InfoANUPQ level 1 reminding you of this should you do this.

After the colon, options a selection of the options listed for the non-interactive PqDescendants function (see PqDescendants (4.4-1)), should be given, separated by commas like record components (see Section Reference: Function Call With Options in the GAP Reference Manual), except that the options SetupFile or PqWorkspace are ignored by the interactive PqDescendants, i.e. the following options are recognised by the interactive PqDescendants function:

• ClassBound := n

• Relators := rels

• OrderBound := n

• StepSize := n, StepSize := list

• RankInitialSegmentSubgroups := n

• SpaceEfficient

• CapableDescendants

• AllDescendants := false

• Exponent := n

• Metabelian

• GroupName := name

• SubList := sub

• BasicAlgorithm

• CustomiseOutput := rec

Notes: The function PqDescendants uses the automorphism group of G which it computes via the package AutPGrp if the automorphism group of G is not already present. If AutPGrp is not installed an error may be raised. If the automorphism group of G is insoluble the pq program will call GAP together with the AutPGrp package for certain orbit-stabilizer calculations.

The attributes and property NuclearRank, MultiplicatorRank and IsCapable are set for each group of the list returned by PqDescendants (see Section Attributes and a Property for fp and pc p-groups).

Let us now repeat the examples previously given for the non-interactive PqDescendants, but this time with the interactive version of PqDescendants:

gap> F := FreeGroup( "a", "b" );; a := F.1;; b := F.2;;
gap> G := PcGroupFpGroup( F / [ a^2, b^2, Comm(b, a) ] );
<pc group of size 4 with 2 generators>
gap> PqStart(G); #This will now be the 4th interactive process running
4
gap> des := PqDescendants( 4 : OrderBound := 6, ClassBound := 5 );;
gap> Length(des);
83
gap> List(des, Size);
[ 8, 8, 8, 16, 16, 16, 32, 16, 16, 16, 16, 16, 32, 32, 64, 64, 32, 32, 32,
32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 32, 32, 32,
64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 32, 32, 32, 32, 64, 64, 64,
64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,
64, 64, 64, 64, 64, 64, 64 ]
gap> List(des, d -> Length( PCentralSeries( d, 2 ) ) - 1 );
[ 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4,
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
4, 4, 4, 5, 5, 5, 5, 5 ]


In the second example we compute all capable descendants of order 27 of the elementary abelian group of order 9.

gap> F := FreeGroup( 2, "g" );;
gap> G := PcGroupFpGroup( F / [ F.1^3, F.2^3, Comm(F.1, F.2) ] );
<pc group of size 9 with 2 generators>
gap> PqStart(G); #This will now be the 5th interactive process running
5
gap> des := PqDescendants( 5 : OrderBound := 3, ClassBound := 2,
>                              CapableDescendants );
[ <pc group of size 27 with 3 generators>,
<pc group of size 27 with 3 generators> ]
gap> List(des, d -> Length( PCentralSeries( d, 3 ) ) - 1 );
[ 2, 2 ]
gap> # For comparison let us now compute all descendants
gap> # (using the non-interactive Pq function)
gap> PqDescendants( G : OrderBound := 3, ClassBound := 2);
[ <pc group of size 27 with 3 generators>,
<pc group of size 27 with 3 generators>,
<pc group of size 27 with 3 generators> ]


In the third example, we compute all capable descendants of the elementary abelian group of order 5^2 which have exponent-5 class at most 3, exponent 5, and are metabelian.

gap> F := FreeGroup( 2, "g" );;
gap> G := PcGroupFpGroup( F / [ F.1^5, F.2^5, Comm(F.2, F.1) ] );
<pc group of size 25 with 2 generators>
gap> PqStart(G); #This will now be the 6th interactive process running
6
gap> des := PqDescendants( 6 : Metabelian, ClassBound := 3,
>                              Exponent := 5, CapableDescendants );
[ <pc group of size 125 with 3 generators>,
<pc group of size 625 with 4 generators>,
<pc group of size 3125 with 5 generators> ]
gap> List(des, d -> Length( PCentralSeries( d, 5 ) ) - 1 );
[ 2, 3, 3 ]
gap> List(des, d -> Length( DerivedSeries( d ) ) );
[ 3, 3, 3 ]
gap> List(des, d -> Maximum( List( Elements(d), Order ) ) );
[ 5, 5, 5 ]


##### 5.3-7 PqSetPQuotientToGroup
 ‣ PqSetPQuotientToGroup( i ) ( function )
 ‣ PqSetPQuotientToGroup( ) ( function )

for the ith or default interactive ANUPQ process, set the p-quotient previously computed by the interactive Pq function (see Pq (5.3-1)) to be the group of the process. This function is supplied to enable the computation of descendants of a p-quotient that is already known to the pq program, via the interactive PqDescendants function (see PqDescendants (5.3-6)), thus avoiding the need to re-submit it and have the pq program recompute it.

Note: See the function PqPGSetDescendantToPcp (PqPGSetDescendantToPcp (5.9-4)) for a mechanism to make (the p-cover of) a particular descendants the current group of the process.

The following example of the usage of PqSetPQuotientToGroup, which is essentially equivalent to what is obtained by running PqExample("PqDescendants-1-i");, redoes the first example of PqDescendants (5.3-6) (which computes the descendants of the Klein four group).

gap> F := FreeGroup( "a", "b" );
<free group on the generators [ a, b ]>
gap> procId := PqStart( F : Prime := 2 );
7
gap> Pq( procId : ClassBound := 1 );
<pc group of size 4 with 2 generators>
gap> PqSetPQuotientToGroup( procId );
gap> des := PqDescendants( procId : OrderBound := 6, ClassBound := 5 );;
gap> Length(des);
83
gap> List(des, Size);
[ 8, 8, 8, 16, 16, 16, 32, 16, 16, 16, 16, 16, 32, 32, 64, 64, 32, 32, 32,
32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 32, 32, 32,
64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 32, 32, 32, 32, 64, 64, 64,
64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,
64, 64, 64, 64, 64, 64, 64 ]
gap> List(des, d -> Length( PCentralSeries( d, 2 ) ) - 1 );
[ 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4,
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
4, 4, 4, 5, 5, 5, 5, 5 ]


#### 5.4 Low-level Interactive ANUPQ functions based on menu items of the pq program

The pq program has 5 menus, the details of which the reader will not normally need to know, but if she wishes to know the details they may be found in the standalone manual: guide.dvi. Both guide.dvi and the pq program refer to the items of these 5 menus as "options", which do not correspond in any way to the options used by any of the GAP functions that interface with the pq program.

Warning: The commands provided in this section are intended to provide something like the interactive functionality one has when running the standalone, from within GAP. The pq standalone (in particular, its "advanced" menus) assumes some expertise of the user; doing the "wrong" thing can cause the program to crash. While a number of safeguards have been provided in the GAP interface to the pq program, these are not foolproof, and the user should exercise care and ensure pre-requisites of the various commands are met.

#### 5.5 General commands

The following commands either use a menu item from whatever menu is "current" for the pq program, or have general application and are not associated with just one menu item of the pq program.

##### 5.5-1 PqNrPcGenerators
 ‣ PqNrPcGenerators( i ) ( function )
 ‣ PqNrPcGenerators( ) ( function )

for the ith or default interactive ANUPQ process, return the number of pc generators of the lower exponent p-class quotient of the group currently determined by the process. This also applies if the pc presentation is not consistent.

##### 5.5-2 PqFactoredOrder
 ‣ PqFactoredOrder( i ) ( function )
 ‣ PqFactoredOrder( ) ( function )

for the ith or default interactive ANUPQ process, return an integer pair [p, n] where p is a prime and n is the number of pc generators (see PqNrPcGenerators (5.5-1)) in the pc presentation of the quotient group currently determined by the process. If this presentation is consistent, then p^n is the order of the quotient group. Otherwise (if tails have been added but the necessary consistency checks, relation collections, exponent law checks and redundant generator eliminations have not yet been done), p^n is an upper bound for the order of the group.

##### 5.5-3 PqOrder
 ‣ PqOrder( i ) ( function )
 ‣ PqOrder( ) ( function )

for the ith or default interactive ANUPQ process, return p^n where [p, n] is the pair as returned by PqFactoredOrder (see PqFactoredOrder (5.5-2)).

##### 5.5-4 PqPClass
 ‣ PqPClass( i ) ( function )
 ‣ PqPClass( ) ( function )

for the ith or default interactive ANUPQ process, return the lower exponent p-class of the quotient group currently determined by the process.

##### 5.5-5 PqWeight
 ‣ PqWeight( i, j ) ( function )
 ‣ PqWeight( j ) ( function )

for the ith or default interactive ANUPQ process, return the weight of the jth pc generator of the lower exponent p-class quotient of the group currently determined by the process, or fail if there is no such numbered pc generator.

##### 5.5-6 PqCurrentGroup
 ‣ PqCurrentGroup( i ) ( function )
 ‣ PqCurrentGroup( ) ( function )

for the ith or default interactive ANUPQ process, return the group whose pc presentation is determined by the process as a GAP pc group (either a lower exponent p-class quotient of the start group or the p-cover of such a quotient).

Notes: See Section Attributes and a Property for fp and pc p-groups for the attributes and property NuclearRank, MultiplicatorRank and IsCapable which may be applied to the group returned by PqCurrentGroup.

##### 5.5-7 PqDisplayPcPresentation
 ‣ PqDisplayPcPresentation( i: [OutputLevel := lev] ) ( function )
 ‣ PqDisplayPcPresentation( : [OutputLevel := lev] ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to display the pc presentation of the lower exponent p-class quotient of the group currently determined by the process.

Except if the last command communicating with the pq program was a p-group generation command (for which there is only a verbose output level), to set the amount of information this command displays you may wish to call PqSetOutputLevel first (see PqSetOutputLevel (5.5-8)), or equivalently pass the option OutputLevel (see 6.2).

Note: For those familiar with the pq program, PqDisplayPcPresentation performs menu item 4 of the current menu of the pq program.

##### 5.5-8 PqSetOutputLevel
 ‣ PqSetOutputLevel( i, lev ) ( function )
 ‣ PqSetOutputLevel( lev ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to set the output level of the pq program to lev.

Note: For those familiar with the pq program, PqSetOutputLevel performs menu item 5 of the main (or advanced) p-Quotient menu, or the Standard Presentation menu.

##### 5.5-9 PqEvaluateIdentities
 ‣ PqEvaluateIdentities( i: [Identities := funcs] ) ( function )
 ‣ PqEvaluateIdentities( : [Identities := funcs] ) ( function )

for the ith or default interactive ANUPQ process, invoke the evaluation of identities defined by the Identities option, and eliminate any redundant pc generators formed. Since a previous value of Identities is saved in the data record of the process, it is unnecessary to pass the Identities if set previously.

Note: This function is mainly implemented at the GAP level. It does not correspond to a menu item of the pq program.

#### 5.6 Commands from the Main p-Quotient menu

##### 5.6-1 PqPcPresentation
 ‣ PqPcPresentation( i: options ) ( function )
 ‣ PqPcPresentation( : options ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to compute the pc presentation of the quotient (determined by options) of the group of the process, which for process i is stored as ANUPQData.io[i].group.

The possible options are the same as for the interactive Pq (see Pq (5.3-1)) function, except for RedoPcp (which, in any case, would be superfluous), namely: Prime, ClassBound, Exponent, Relators, GroupName, Metabelian, Identities and OutputLevel (see Chapter ANUPQ Options for a detailed description for these options). The option Prime is required unless already provided to PqStart.

Notes

The pc presentation is held by the pq program. In contrast to Pq (see Pq (5.3-1)), no GAP pc group is returned; see PqCurrentGroup (PqCurrentGroup (5.5-6)) if you need the corresponding GAP pc group.

PqPcPresentation(i: options); is roughly equivalent to the following sequence of low-level commands:

PqPcPresentation(i: opts); #class 1 call
for c in [2 .. class] do
PqNextClass(i);
od;


where opts is options except with the ClassBound option set to 1, and class is either the maximum class of a p-quotient of the group of the process or the user-supplied value of the option ClassBound (whichever is smaller). If the Identities option has been set, both the first PqPcPresentation class 1 call and the PqNextClass calls invoke PqEvaluateIdentities(i); as their final step.

For those familiar with the pq program, PqPcPresentation performs menu item 1 of the main p-Quotient menu.

##### 5.6-2 PqSavePcPresentation
 ‣ PqSavePcPresentation( i, filename ) ( function )
 ‣ PqSavePcPresentation( filename ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to save the pc presentation previously computed for the quotient of the group of that process to the file with name filename. If the first character of the string filename is not /, filename is assumed to be the path of a writable file relative to the directory in which GAP was started. A saved file may be restored by PqRestorePcPresentation (see PqRestorePcPresentation (5.6-3)).

Note: For those familiar with the pq program, PqSavePcPresentation performs menu item 2 of the main p-Quotient menu.

##### 5.6-3 PqRestorePcPresentation
 ‣ PqRestorePcPresentation( i, filename ) ( function )
 ‣ PqRestorePcPresentation( filename ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to restore the pc presentation previously saved to filename, by PqSavePcPresentation (see PqSavePcPresentation (5.6-2)). If the first character of the string filename is not /, filename is assumed to be the path of a readable file relative to the directory in which GAP was started.

Note: For those familiar with the pq program, PqRestorePcPresentation performs menu item 3 of the main p-Quotient menu.

##### 5.6-4 PqNextClass
 ‣ PqNextClass( i: [QueueFactor] ) ( function )
 ‣ PqNextClass( : [QueueFactor] ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to calculate the next class of ANUPQData.io[i].group.

PqNextClass accepts the option QueueFactor (see also 6.2) which should be a positive integer if automorphisms have been previously supplied. If the pq program requires a queue factor and none is supplied via the option QueueFactor a default of 15 is taken.

Notes

The single command: PqNextClass(i); is equivalent to executing

PqComputePCover(i);
PqCollectDefiningRelations(i);
PqDoExponentChecks(i);
PqEliminateRedundantGenerators(i);


If the Identities option is set the PqEliminateRedundantGenerators(i); step is essentially replaced by PqEvaluateIdentities(i); (which invokes its own elimination of redundant generators).

For those familiar with the pq program, PqNextClass performs menu item 6 of the main p-Quotient menu.

##### 5.6-5 PqComputePCover
 ‣ PqComputePCover( i ) ( function )
 ‣ PqComputePCover( ) ( function )

for the ith or default interactive ANUPQ processi, directi, the pq program to compute the p-covering group of ANUPQData.io[i].group. In contrast to the function PqPCover (see PqPCover (4.1-3)), this function does not return a GAP pc group.

Notes

The single command: PqComputePCover(i); is equivalent to executing

PqSetupTablesForNextClass(i);
PqTails(i, 0);
PqDoConsistencyChecks(i, 0, 0);
PqEliminateRedundantGenerators(i);


For those familiar with the pq program, PqComputePCover performs menu item 7 of the main p-Quotient menu.

##### 5.7-1 PqCollect
 ‣ PqCollect( i, word ) ( function )
 ‣ PqCollect( word ) ( function )

for the ith or default interactive ANUPQ process, instruct the pq program to do a collection on word, a word in the current pc generators (the form of word required is described below). PqCollect returns the resulting word of the collection as a list of generator number, exponent pairs (the same form as the second allowed input form of word; see below).

The argument word may be input in either of the following ways:

1. word may be a string, where the ith pc generator is represented by xi, e.g. "x3*x2^2*x1". This way is quite versatile as parentheses and left-normed commutators -- using square brackets, in the same way as PqGAPRelators (see PqGAPRelators (3.4-2)) -- are permitted; word is checked for correct syntax via PqParseWord (see PqParseWord (3.4-3)).

2. Otherwise, word must be a list of generator number, exponent pairs of integers, i.e.  each pair represents a "syllable" so that [ [3, 1], [2, 2], [1, 1] ] represents the same word as that of the example given for the first allowed form of word.

Note: For those familiar with the pq program, PqCollect performs menu item 1 of the Advanced p-Quotient menu.

##### 5.7-2 PqSolveEquation
 ‣ PqSolveEquation( i, a, b ) ( function )
 ‣ PqSolveEquation( a, b ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to solve a * x = b for x, where a and b are words in the pc generators. For the representation of these words see the description of the function PqCollect (PqCollect (5.7-1)).

Note: For those familiar with the pq program, PqSolveEquation performs menu item 2 of the Advanced p-Quotient menu.

##### 5.7-3 PqCommutator
 ‣ PqCommutator( i, words, pow ) ( function )
 ‣ PqCommutator( words, pow ) ( function )

for the ith or default interactive ANUPQ process, instruct the pq program to compute the left normed commutator of the list words of words in the current pc generators raised to the integer power pow, and return the resulting word as a list of generator number, exponent pairs. The form required for each word of words is the same as that required for the word argument of PqCollect (see PqCollect (5.7-1)). The form of the output word is also the same as for PqCollect.

Note: For those familiar with the pq program, PqCommutator performs menu item 3 of the Advanced p-Quotient menu.

##### 5.7-4 PqSetupTablesForNextClass
 ‣ PqSetupTablesForNextClass( i ) ( function )
 ‣ PqSetupTablesForNextClass( ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to set up tables for the next class. As as side-effect, after PqSetupTablesForNextClass(i) the value returned by PqPClass(i) will be one more than it was previously.

Note: For those familiar with the pq program, PqSetupTablesForNextClass performs menu item 6 of the Advanced p-Quotient menu.

##### 5.7-5 PqTails
 ‣ PqTails( i, weight ) ( function )
 ‣ PqTails( weight ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to compute and add tails of weight weight if weight is in the integer range [2 .. PqPClass(i)] (assuming i is the number of the process, even in the default case) or for all weights if weight = 0.

If weight is non-zero, then tails that introduce new generators for only weight weight are computed and added, and in this case and if weight < PqPClass(i), it is assumed that the tails that introduce new generators for each weight from PqPClass(i) down to weight weight + 1 have already been added. You may wish to call PqSetMetabelian (see PqSetMetabelian (5.7-16)) prior to calling PqTails.

Notes

For its use in the context of finding the next class see PqNextClass (5.6-4); in particular, a call to PqSetupTablesForNextClass (see PqSetupTablesForNextClass (5.7-4)) needs to have been made prior to calling PqTails.

The single command: PqTails(i, weight); is equivalent to

PqComputeTails(i, weight);


For those familiar with the pq program, PqTails uses menu item 7 of the Advanced p-Quotient menu.

##### 5.7-6 PqComputeTails
 ‣ PqComputeTails( i, weight ) ( function )
 ‣ PqComputeTails( weight ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to compute tails of weight weight if weight is in the integer range [2 .. PqPClass(i)] (assuming i is the number of the process, even in the default case) or for all weights if weight = 0. See PqTails (PqTails (5.7-5)) for more details.

Note: For those familiar with the pq program, PqComputeTails uses menu item 7 of the Advanced p-Quotient menu.

 ‣ PqAddTails( i, weight ) ( function )
 ‣ PqAddTails( weight ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to add the tails of weight weight, previously computed by PqComputeTails (see PqComputeTails (5.7-6)), if weight is in the integer range [2 .. PqPClass(i)] (assuming i is the number of the process, even in the default case) or for all weights if weight = 0. See PqTails (PqTails (5.7-5)) for more details.

Note: For those familiar with the pq program, PqAddTails uses menu item 7 of the Advanced p-Quotient menu.

##### 5.7-8 PqDoConsistencyChecks
 ‣ PqDoConsistencyChecks( i, weight, type ) ( function )
 ‣ PqDoConsistencyChecks( weight, type ) ( function )

for the ith or default interactive ANUPQ process, do consistency checks for weight weight if weight is in the integer range [3 .. PqPClass(i)] (assuming i is the number of the process) or for all weights if weight = 0, and for type type if type is in the range [1, 2, 3] (see below) or for all types if type = 0. (For its use in the context of finding the next class see PqNextClass (5.6-4).)

The type of a consistency check is defined as follows. PqDoConsistencyChecks(i, weight, type) for weight in [3 .. PqPClass(i)] and the given value of type invokes the equivalent of the following PqDoConsistencyCheck calls (see PqDoConsistencyCheck (5.7-17)):

type = 1:

PqDoConsistencyCheck(i, a, a, a) checks 2 * PqWeight(i, a) + 1 = weight, for pc generators of index a.

type = 2:

PqDoConsistencyCheck(i, b, b, a) checks for pc generators of indices b, a satisfyingx both b > a and PqWeight(i, b) + PqWeight(i, a) + 1 = weight.

type = 3:

PqDoConsistencyCheck(i, c, b, a) checks for pc generators of indices c, b, a satisfying c > b > a and the sum of the weights of these generators equals weight.

Notes

PqWeight(i, j) returns the weight of the jth pc generator, for process i (see PqWeight (5.5-5)).

It is assumed that tails for the given weight (or weights) have already been added (see PqTails (5.7-5)).

For those familiar with the pq program, PqDoConsistencyChecks performs menu item 8 of the Advanced p-Quotient menu.

##### 5.7-9 PqCollectDefiningRelations
 ‣ PqCollectDefiningRelations( i ) ( function )
 ‣ PqCollectDefiningRelations( ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to collect the images of the defining relations of the original fp group of the process, with respect to the current pc presentation, in the context of finding the next class (see PqNextClass (5.6-4)). If the tails operation is not complete then the relations may be evaluated incorrectly.

Note: For those familiar with the pq program, PqCollectDefiningRelations performs menu item 9 of the Advanced p-Quotient menu.

##### 5.7-10 PqCollectWordInDefiningGenerators
 ‣ PqCollectWordInDefiningGenerators( i, word ) ( function )
 ‣ PqCollectWordInDefiningGenerators( word ) ( function )

for the ith or default interactive ANUPQ process, take a user-defined word word in the defining generators of the original presentation of the fp or pc group of the process. Each generator is mapped into the current pc presentation, and the resulting word is collected with respect to the current pc presentation. The result of the collection is returned as a list of generator number, exponent pairs.

The word argument may be input in either of the two ways described for PqCollect (see PqCollect (5.7-1)).

Note: For those familiar with the pq program, PqCollectDefiningGenerators performs menu item 23 of the Advanced p-Quotient menu.

##### 5.7-11 PqCommutatorDefiningGenerators
 ‣ PqCommutatorDefiningGenerators( i, words, pow ) ( function )
 ‣ PqCommutatorDefiningGenerators( words, pow ) ( function )

for the ith or default interactive ANUPQ process, take a list words of user-defined words in the defining generators of the original presentation of the fp or pc group of the process, and an integer power pow. Each generator is mapped into the current pc presentation. The list words is interpreted as a left-normed commutator which is then raised to pow and collected with respect to the current pc presentation. The result of the collection is returned as a list of generator number, exponent pairs.

Note For those familiar with the pq program, PqCommutatorDefiningGenerators performs menu item 24 of the Advanced p-Quotient menu.

##### 5.7-12 PqDoExponentChecks
 ‣ PqDoExponentChecks( i: [Bounds := list] ) ( function )
 ‣ PqDoExponentChecks( : [Bounds := list] ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to do exponent checks for weights (inclusively) between the bounds of Bounds or for all weights if Bounds is not given. The value list of Bounds (assuming the interactive process is numbered i) should be a list of two integers low, high satisfying 1 le low le high le PqPClass(i) (see PqPClass (5.5-4)). If no exponent law has been specified, no exponent checks are performed.

Note: For those familiar with the pq program, PqDoExponentChecks performs menu item 10 of the Advanced p-Quotient menu.

##### 5.7-13 PqEliminateRedundantGenerators
 ‣ PqEliminateRedundantGenerators( i ) ( function )
 ‣ PqEliminateRedundantGenerators( ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to eliminate redundant generators of the current p-quotient.

Note: For those familiar with the pq program, PqEliminateRedundantGenerators performs menu item 11 of the Advanced p-Quotient menu.

##### 5.7-14 PqRevertToPreviousClass
 ‣ PqRevertToPreviousClass( i ) ( function )
 ‣ PqRevertToPreviousClass( ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to abandon the current class and revert to the previous class.

Note: For those familiar with the pq program, PqRevertToPreviousClass performs menu item 12 of the Advanced p-Quotient menu.

##### 5.7-15 PqSetMaximalOccurrences
 ‣ PqSetMaximalOccurrences( i, noccur ) ( function )
 ‣ PqSetMaximalOccurrences( noccur ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to set maximal occurrences of the weight 1 generators in the definitions of pcp generators of the group of the process. This can be used to avoid the definition of generators of which one knows for theoretical reasons that they would be eliminated later on.

The argument noccur must be a list of non-negative integers of length the number of weight 1 generators (i.e. the rank of the class 1 p-quotient of the group of the process). An entry of 0 for a particular generator indicates that there is no limit on the number of occurrences for the generator.

Note: For those familiar with the pq program, PqSetMaximalOccurrences performs menu item 13 of the Advanced p-Quotient menu.

##### 5.7-16 PqSetMetabelian
 ‣ PqSetMetabelian( i ) ( function )
 ‣ PqSetMetabelian( ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to enforce metabelian-ness.

Note: For those familiar with the pq program, PqSetMetabelian performs menu item 14 of the Advanced p-Quotient menu.

##### 5.7-17 PqDoConsistencyCheck
 ‣ PqDoConsistencyCheck( i, c, b, a ) ( function )
 ‣ PqDoConsistencyCheck( c, b, a ) ( function )
 ‣ PqJacobi( i, c, b, a ) ( function )
 ‣ PqJacobi( c, b, a ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to do the consistency check for the pc generators with indices c, b, a which should be non-increasing positive integers, i.e. c ge b ge a.

There are 3 types of consistency checks:

\begin{array}{rclrl} (a^n)a &=& a(a^n) && {\rm (Type\ 1)} \\ (b^n)a &=& b^{(n-1)}(ba), b(a^n) = (ba)a^{(n-1)} && {\rm (Type\ 2)} \\ c(ba) &=& (cb)a && {\rm (Type\ 3)} \\ \end{array}

The reason some people talk about Jacobi relations instead of consistency checks becomes clear when one looks at the consistency check of type 3:

\begin{array}{rcl} c(ba) &=& a c[c,a] b[b,a] = acb [c,a][c,a,b][b,a] = \dots \\ (cb)a &=& b c[c,b] a = a b[b,a] c[c,a] [c,b][c,b,a] \\ &=& abc [b,a] [b,a,c] [c,a] [c,b] [c,b,a] = \dots \\ \end{array}

Each collection would normally carry on further. But one can see already that no other commutators of weight 3 will occur. After all terms of weight one and weight two have been moved to the left we end up with:

\begin{array}{rcl} & &abc [b,a] [c,a] [c,b] [c,a,b] \dots \\ &=&abc [b,a] [c,a] [c,b] [c,b,a] [b,a,c] \dots \\ \end{array}

Modulo terms of weight 4 this is equivalent to

[c,a,b] [b,c,a] [a,b,c] = 1

which is the Jacobi identity.

See also PqDoConsistencyChecks (PqDoConsistencyChecks (5.7-8)).

Note: For those familiar with the pq program, PqDoConsistencyCheck and PqJacobi perform menu item 15 of the Advanced p-Quotient menu.

##### 5.7-18 PqCompact
 ‣ PqCompact( i ) ( function )
 ‣ PqCompact( ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to do a compaction of its work space. This function is safe to perform only at certain points in time.

Note: For those familiar with the pq program, PqCompact performs menu item 16 of the Advanced p-Quotient menu.

##### 5.7-19 PqEchelonise
 ‣ PqEchelonise( i ) ( function )
 ‣ PqEchelonise( ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to echelonise the word most recently collected by PqCollect or PqCommutator against the relations of the current pc presentation, and return the number of the generator made redundant or fail if no generator was made redundant. A call to PqCollect (see PqCollect (5.7-1)) or PqCommutator (see PqCommutator (5.7-3)) needs to be performed prior to using this command.

Note: For those familiar with the pq program, PqEchelonise performs menu item 17 of the Advanced p-Quotient menu.

##### 5.7-20 PqSupplyAutomorphisms
 ‣ PqSupplyAutomorphisms( i, mlist ) ( function )
 ‣ PqSupplyAutomorphisms( mlist ) ( function )

for the ith or default interactive ANUPQ process, supply the automorphism data provided by the list mlist of matrices with non-negative integer coefficients. Each matrix in mlist describes one automorphism in the following way.

• The rows of each matrix correspond to the pc generators of weight one.

• Each row is the exponent vector of the image of the corresponding weight one generator under the respective automorphism.

Note: For those familiar with the pq program, PqSupplyAutomorphisms uses menu item 18 of the Advanced p-Quotient menu.

##### 5.7-21 PqExtendAutomorphisms
 ‣ PqExtendAutomorphisms( i ) ( function )
 ‣ PqExtendAutomorphisms( ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to extend automorphisms of the p-quotient of the previous class to the p-quotient of the present class.

Note: For those familiar with the pq program, PqExtendAutomorphisms uses menu item 18 of the Advanced p-Quotient menu.

##### 5.7-22 PqApplyAutomorphisms
 ‣ PqApplyAutomorphisms( i, qfac ) ( function )
 ‣ PqApplyAutomorphisms( qfac ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to apply automorphisms; qfac is the queue factor e.g. 15.

Note: For those familiar with the pq program, PqCloseRelations performs menu item 19 of the Advanced p-Quotient menu.

##### 5.7-23 PqDisplayStructure
 ‣ PqDisplayStructure( i: [Bounds := list] ) ( function )
 ‣ PqDisplayStructure( : [Bounds := list] ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to display the structure for the pcp generators numbered (inclusively) between the bounds of Bounds or for all generators if Bounds is not given. The value list of Bounds (assuming the interactive process is numbered i) should be a list of two integers low, high satisfying 1 le low le high le PqNrPcGenerators(i) (see PqNrPcGenerators (5.5-1)). PqDisplayStructure also accepts the option OutputLevel (see 6.2).

Explanation of output

New generators are defined as commutators of previous generators and generators of class 1 or as p-th powers of generators that have themselves been defined as p-th powers. A generator is never defined as p-th power of a commutator.

Therefore, there are two cases: all the numbers on the righthand side are either the same or they differ. Below, gi refers to the ith defining generator.

• If the righthand side numbers are all the same, then the generator is a p-th power (of a p-th power of a p-th power, etc.). The number of repeated digits say how often a p-th power has to be taken.

In the following example, the generator number 31 is the eleventh power of generator 17 which in turn is an eleventh power and so on:

\begintt #I 31 is defined on 17^11 = 1 1 1 1 1 \endtt So generator 31 is obtained by taking the eleventh power of generator 1 five times.

• If the numbers are not all the same, the generator is defined by a commutator. If the first two generator numbers differ, the generator is defined as a left-normed commutator of the weight one generators, e.g.

\begintt #I 19 is defined on [11, 1] = 2 1 1 1 1 \endtt Here, generator 19 is defined as the commutator of generator 11 and generator 1 which is the same as the left-normed commutator [x2, x1, x1, x1, x1]. One can check this by tracing back the definition of generator 11 until one gets to a generator of class 1.

• If the first two generator numbers are identical, then the left most component of the left-normed commutator is a p-th power, e.g.

\begintt #I 25 is defined on [14, 1] = 1 1 2 1 1 \endtt

In this example, generator 25 is defined as commutator of generator 14 and generator 1. The left-normed commutator is

[(x1^{11})^{11}, x2, x1, x1]

Again, this can be verified by tracing back the definitions.

Note: For those familiar with the pq program, PqDisplayStructure performs menu item 20 of the Advanced p-Quotient menu.

##### 5.7-24 PqDisplayAutomorphisms
 ‣ PqDisplayAutomorphisms( i: [Bounds := list] ) ( function )
 ‣ PqDisplayAutomorphisms( : [Bounds := list] ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to display the automorphism actions on the pcp generators numbered (inclusively) between the bounds of Bounds or for all generators if Bounds is not given. The value list of Bounds (assuming the interactive process is numbered i) should be a list of two integers low, high satisfying 1 le low le high le PqNrPcGenerators(i) (see PqNrPcGenerators (5.5-1)). PqDisplayStructure also accepts the option OutputLevel (see 6.2).

Note: For those familiar with the pq program, PqDisplayAutomorphisms performs menu item 21 of the Advanced p-Quotient menu.

##### 5.7-25 PqWritePcPresentation
 ‣ PqWritePcPresentation( i, filename ) ( function )
 ‣ PqWritePcPresentation( filename ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to write a pc presentation of a previously-computed quotient of the group of that process, to the file with name filename. Here the group of a process is the one given as first argument when PqStart was called to initiate that process (for process i the group is stored as ANUPQData.io[i].group). If the first character of the string filename is not /, filename is assumed to be the path of a writable file relative to the directory in which GAP was started. If a pc presentation has not been previously computed by the pq program, then pq is called to compute it first, effectively invoking PqPcPresentation (see PqPcPresentation (5.6-1)).

Note: For those familiar with the pq program, PqPcWritePresentation performs menu item 25 of the Advanced p-Quotient menu.

#### 5.8 Commands from the Standard Presentation menu

##### 5.8-1 PqSPComputePcpAndPCover
 ‣ PqSPComputePcpAndPCover( i: options ) ( function )
 ‣ PqSPComputePcpAndPCover( : options ) ( function )

for the ith or default interactive ANUPQ process, directs the pq program to compute for the group of that process a pc presentation up to the p-quotient of maximum class or the value of the option ClassBound and the p-cover of that quotient, and sets up tabular information required for computation of a standard presentation. Here the group of a process is the one given as first argument when PqStart was called to initiate that process (for process i the group is stored as ANUPQData.io[i].group).

The possible options are Prime, ClassBound, Relators, Exponent, Metabelian and OutputLevel (see Chapter ANUPQ Options for detailed descriptions of these options). The option Prime is normally determined via PrimePGroup, and so is not required unless the group doesn't know it's a p-group and HasPrimePGroup returns false.

Note: For those familiar with the pq program, PqSPComputePcpAndPCover performs option 1 of the Standard Presentation menu.

##### 5.8-2 PqSPStandardPresentation
 ‣ PqSPStandardPresentation( i[, mlist]: [options] ) ( function )
 ‣ PqSPStandardPresentation( [mlist]: [options] ) ( function )

for the ith or default interactive ANUPQ process, inputs data given by options to compute a standard presentation for the group of that process. If argument mlist is given it is assumed to be the automorphism group data required. Otherwise it is assumed that a call to either Pq (see Pq (5.3-1)) or PqEpimorphism (see PqEpimorphism (5.3-2)) has generated a p-quotient and that GAP can compute its automorphism group from which the necessary automorphism group data can be derived. The group of the process is the one given as first argument when PqStart was called to initiate the process (for process i the group is stored as ANUPQData.io[i].group and the p-quotient if existent is stored as ANUPQData.io[i].pQuotient). If mlist is not given and a p-quotient of the group has not been previously computed a class 1 p-quotient is computed.

PqSPStandardPresentation accepts three options, all optional:

• ClassBound := n

• PcgsAutomorphisms

• StandardPresentationFile := filename

If ClassBound is omitted it defaults to 63.

Detailed descriptions of the above options may be found in Chapter ANUPQ Options.

Note: For those familiar with the pq program, PqSPPcPresentation performs menu item 2 of the Standard Presentation menu.

##### 5.8-3 PqSPSavePresentation
 ‣ PqSPSavePresentation( i, filename ) ( function )
 ‣ PqSPSavePresentation( filename ) ( function )

for the ith or default interactive ANUPQ process, directs the pq program to save the standard presentation previously computed for the group of that process to the file with name filename, where the group of a process is the one given as first argument when PqStart was called to initiate that process. If the first character of the string filename is not /, filename is assumed to be the path of a writable file relative to the directory in which GAP was started.

Note: For those familiar with the pq program, PqSPSavePresentation performs menu item 3 of the Standard Presentation menu.

##### 5.8-4 PqSPCompareTwoFilePresentations
 ‣ PqSPCompareTwoFilePresentations( i, f1, f2 ) ( function )
 ‣ PqSPCompareTwoFilePresentations( f1, f2 ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to compare the presentations in the files with names f1 and f2 and returns true if they are identical and false otherwise. For each of the strings f1 and f2, if the first character is not a / then it is assumed to be the path of a readable file relative to the directory in which GAP was started.

Notes

The presentations in files f1 and f2 must have been generated by the pq program but they do not need to be standard presentations. If If the presentations in files f1 and f2 have been generated by PqSPStandardPresentation (see PqSPStandardPresentation (5.8-2)) then a false response from PqSPCompareTwoFilePresentations says the groups defined by those presentations are not isomorphic.

For those familiar with the pq program, PqSPCompareTwoFilePresentations performs menu item 6 of the Standard Presentation menu.

##### 5.8-5 PqSPIsomorphism
 ‣ PqSPIsomorphism( i ) ( function )
 ‣ PqSPIsomorphism( ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to compute the isomorphism mapping from the p-group of the process to its standard presentation. This function provides a description only; for a GAP object, use EpimorphismStandardPresentation (see EpimorphismStandardPresentation (5.3-5)).

Note: For those familiar with the pq program, PqSPIsomorphism performs menu item 8 of the Standard Presentation menu.

#### 5.9 Commands from the Main p-Group Generation menu

Note that the p-group generation commands can only be applied once the pq program has produced a pc presentation of some quotient group of the "group of the process".

##### 5.9-1 PqPGSupplyAutomorphisms
 ‣ PqPGSupplyAutomorphisms( i[, mlist]: options ) ( function )
 ‣ PqPGSupplyAutomorphisms( [mlist]: options ) ( function )

for the ith or default interactive ANUPQ process, supply the pq program with the automorphism group data needed for the current quotient of the group of that process (for process i the group is stored as ANUPQData.io[i].group). For a description of the format of mlist see PqSupplyAutomorphisms (5.7-20). The options possible are NumberOfSolubleAutomorphisms and RelativeOrders. (Detailed descriptions of these options may be found in Chapter ANUPQ Options.)

If mlist is omitted, the automorphism data is determined from the group of the process which must have been a p-group in pc presentation.

Note: For those familiar with the pq program, PqPGSupplyAutomorphisms performs menu item 1 of the main p-Group Generation menu.

##### 5.9-2 PqPGExtendAutomorphisms
 ‣ PqPGExtendAutomorphisms( i ) ( function )
 ‣ PqPGExtendAutomorphisms( ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to compute the extensions of the automorphisms of the p-quotient of the previous class to the p-quotient of the current class. You may wish to set the InfoLevel of InfoANUPQ to 2 (or more) in order to see the output from the pq program (see InfoANUPQ (3.3-1)).

Note: For those familiar with the pq program, PqPGExtendAutomorphisms performs menu item 2 of the main or advanced p-Group Generation menu.

##### 5.9-3 PqPGConstructDescendants
 ‣ PqPGConstructDescendants( i: options ) ( function )
 ‣ PqPGConstructDescendants( : options ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to construct descendants prescribed by options, and return the number of descendants constructed (compare function PqDescendants (4.4-1) which returns the list of descendants). The options possible are ClassBound, OrderBound, StepSize, PcgsAutomorphisms, RankInitialSegmentSubgroups, SpaceEfficient, CapableDescendants, AllDescendants, Exponent, Metabelian, BasicAlgorithm, CustomiseOutput. (Detailed descriptions of these options may be found in Chapter ANUPQ Options.)

PqPGConstructDescendants requires that the pq program has previously computed a pc presentation and a p-cover for a p-quotient of some class of the group of the process.

Note: For those familiar with the pq program, PqPGConstructDescendants performs menu item 5 of the main p-Group Generation menu.

##### 5.9-4 PqPGSetDescendantToPcp
 ‣ PqPGSetDescendantToPcp( i, cls, n ) ( function )
 ‣ PqPGSetDescendantToPcp( cls, n ) ( function )
 ‣ PqPGSetDescendantToPcp( i: [Filename := name] ) ( function )
 ‣ PqPGSetDescendantToPcp( : [Filename := name] ) ( function )
 ‣ PqPGRestoreDescendantFromFile( i, cls, n ) ( function )
 ‣ PqPGRestoreDescendantFromFile( cls, n ) ( function )
 ‣ PqPGRestoreDescendantFromFile( i: [Filename := name] ) ( function )
 ‣ PqPGRestoreDescendantFromFile( : [Filename := name] ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to restore group n of class cls from a temporary file, where cls and n are positive integers, or the group stored in name. PqPGSetDescendantToPcp and PqPGRestoreDescendantFromFile are synonyms; they make sense only after a prior call to construct descendants by say PqPGConstructDescendants (see PqPGConstructDescendants (5.9-3)) or the interactive PqDescendants (see PqDescendants (5.3-6)). In the Filename option forms, the option defaults to the last filename in which a presentation was stored by the pq program.

Notes

Since the PqPGSetDescendantToPcp and PqPGRestoreDescendantFromFile are intended to be used in calculation of further descendants the pq program computes the p-cover of the restored descendant. Hence, PqCurrentGroup used immediately after one of these commands returns the p-cover of the restored descendant rather than the descendant itself.

For those familiar with the pq program, PqPGSetDescendantToPcp and PqPGRestoreDescendantFromFile perform menu item 3 of the main or advanced p-Group Generation menu.

The functions below perform the component algorithms of PqPGConstructDescendants (see PqPGConstructDescendants (5.9-3)). You can get some idea of their usage by trying PqExample("Nott-APG-Rel-i");. You can get some idea of the breakdown of PqPGConstructDescendants into these functions by comparing the previous output with PqExample("Nott-PG-Rel-i");.

These functions are intended for use only by "experts"; please contact the authors of the package if you genuinely have a need for them and need any amplified descriptions.

##### 5.10-1 PqAPGDegree
 ‣ PqAPGDegree( i, step, rank: [Exponent := n] ) ( function )
 ‣ PqAPGDegree( step, rank: [Exponent := n] ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to invoke menu item 6 of the Advanced p-Group Generation menu. Here the step-size step and the rank rank are positive integers and are the arguments required by the pq program. See 6.2 for the one recognised option Exponent.

##### 5.10-2 PqAPGPermutations
 ‣ PqAPGPermutations( i: options ) ( function )
 ‣ PqAPGPermutations( : options ) ( function )

for the ith or default interactive ANUPQ process, direct the pq program to perform menu item 7 of the Advanced p-Group Generation menu. Here the options options recognised are PcgsAutomorphisms, SpaceEfficient, PrintAutomorphisms and PrintPermutations (see Chapter ANUPQ Options for details).

##### 5.10-3 PqAPGOrbits
 ‣ PqAPGOrbits( i: options ) ( function )
 ‣ PqAPGOrbits( : options ) ( function )

for the ith or default interactive ANUPQ process, direct the pq to perform menu item 8 of the Advanced p-Group Generation menu.

Here the options options recognised are PcgsAutomorphisms, SpaceEfficient and CustomiseOutput (see Chapter ANUPQ Options for details). For the CustomiseOutput option only the setting of the orbit is recognised (all other fields if set are ignored).

##### 5.10-4 PqAPGOrbitRepresentatives
 ‣ PqAPGOrbitRepresentatives( i: options ) ( function )
 ‣ PqAPGOrbitRepresentatives( : options ) ( function )

for the ith or default interactive ANUPQ process, direct the pq to perform item 9 of the Advanced p-Group Generation menu.

The options options may be any selection of the following: PcgsAutomorphisms, SpaceEfficient, Exponent, Metabelian, CapableDescendants (or AllDescendants), CustomiseOutput (where only the group and autgroup fields are recognised) and Filename (see Chapter ANUPQ Options for details). If Filename is omitted the reduced p-cover is written to the file "redPCover" in the temporary directory whose name is stored in ANUPQData.tmpdir.

##### 5.10-5 PqAPGSingleStage
 ‣ PqAPGSingleStage( i: options ) ( function )
 ‣ PqAPGSingleStage( : options ) ( function )

for the ith or default interactive ANUPQ process, direct the pq to perform option 5 of the Advanced p-Group Generation menu.

The possible options are StepSize, PcgsAutomorphisms, RankInitialSegmentSubgroups, SpaceEfficient, CapableDescendants, AllDescendants, Exponent, Metabelian, BasicAlgorithm and CustomiseOutput. (Detailed descriptions of these options may be found in Chapter ANUPQ Options.)

#### 5.11 Primitive Interactive ANUPQ Process Read/Write Functions

For those familiar with using the pq program as a standalone we provide primitive read/write tools to communicate directly with an interactive ANUPQ process, started via PqStart. For the most part, it is up to the user to translate the output strings from pq program into a form useful in GAP.

 ‣ PqRead( i ) ( function )
 ‣ PqRead( ) ( function )

read a complete line of ANUPQ output, from the ith or default interactive ANUPQ process, if there is output to be read and returns fail otherwise. When successful, the line is returned as a string complete with trailing newline, colon, or question-mark character. Please note that it is possible to be "too quick" (i.e. the return can be fail purely because the output from ANUPQ is not there yet), but if PqRead finds any output at all, it waits for a complete line. PqRead also writes the line read via Info at InfoANUPQ level 2. It doesn't try to distinguish banner and menu output from other output of the pq program.

 ‣ PqReadAll( i ) ( function )
 ‣ PqReadAll( ) ( function )

read and return as many complete lines of ANUPQ output, from the ith or default interactive ANUPQ process, as there are to be read, at the time of the call, as a list of strings with any trailing newlines removed and returns the empty list otherwise. PqReadAll also writes each line read via Info at InfoANUPQ level 2. It doesn't try to distinguish banner and menu output from other output of the pq program. Whenever PqReadAll finds only a partial line, it waits for the complete line, thus increasing the probability that it has captured all the output to be had from ANUPQ.

 ‣ PqReadUntil( i, IsMyLine ) ( function )
 ‣ PqReadUntil( IsMyLine ) ( function )
 ‣ PqReadUntil( i, IsMyLine, Modify ) ( function )
 ‣ PqReadUntil( IsMyLine, Modify ) ( function )

read complete lines of ANUPQ output, from the ith or default interactive ANUPQ process, "chomps" them (i.e. removes any trailing newline character), emits them to Info at InfoANUPQ level 2 (without trying to distinguish banner and menu output from other output of the pq program), and applies the function Modify (where Modify is just the identity map/function for the first two forms) until a "chomped" line line for which IsMyLine( Modify(line) ) is true. PqReadUntil returns the list of Modify-ed "chomped" lines read.

Notes: When provided by the user, Modify should be a function that accepts a single string argument.

IsMyLine should be a function that is able to accept the output of Modify (or take a single string argument when Modify is not provided) and should return a boolean.

If IsMyLine( Modify(line) ) is never true, PqReadUntil will wait indefinitely.

##### 5.11-4 PqWrite
 ‣ PqWrite( i, string ) ( function )
 ‣ PqWrite( string ) ( function )

write string to the ith or default interactive ANUPQ process; string must be in exactly the form the ANUPQ standalone expects. The command is echoed via Info at InfoANUPQ level 3 (with a "ToPQ> " prompt); i.e. do SetInfoLevel(InfoANUPQ, 3); to see what is transmitted to the pq program. PqWrite returns true if successful in writing to the stream of the interactive ANUPQ process, and fail otherwise.

Note: If PqWrite returns fail it means that the ANUPQ process has died.

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