In this chapter we describe in detail all the options used by functions of the ANUPQ package. Note that by "options" we mean GAP options that are passed to functions after the arguments and separated from the arguments by a colon as described in Chapter Reference: Function Calls in the Reference Manual. The user is strongly advised to read Section Hints and Warnings regarding the use of Options.
‣ AllANUPQoptions( ) | ( function ) |
lists all the GAP options defined for functions of the ANUPQ package:
gap> AllANUPQoptions(); [ "AllDescendants", "BasicAlgorithm", "Bounds", "CapableDescendants", "ClassBound", "CustomiseOutput", "Exponent", "Filename", "GroupName", "Identities", "Metabelian", "NumberOfSolubleAutomorphisms", "OrderBound", "OutputLevel", "PcgsAutomorphisms", "PqWorkspace", "Prime", "PrintAutomorphisms", "PrintPermutations", "QueueFactor", "RankInitialSegmentSubgroups", "RedoPcp", "RelativeOrders", "Relators", "SetupFile", "SpaceEfficient", "StandardPresentationFile", "StepSize", "SubList", "TreeDepth", "pQuotient" ]
The following global variable gives a partial breakdown of where the above options are used.
‣ ANUPQoptions | ( global variable ) |
is a record of lists of names of admissible ANUPQ options, such that each field is either the name of a "key" ANUPQ function or other (for a miscellaneous list of functions) and the corresponding value is the list of option names that are admissible for the function (or miscellaneous list of functions).
Also, from within a GAP session, you may use GAP's help browser (see Chapter Reference: The Help System in the GAP Reference Manual); to find out about any particular ANUPQ option, simply type: "?option option", where option is one of the options listed above without any quotes, e.g.
gap> ?option Prime
will display the sections in this manual that describe the Prime option. In fact the first 4 are for the functions that have Prime as an option and the last actually describes the option. So follow up by choosing
gap> ?5
This is also the pattern for other options (the last section of the list always describes the option; the other sections are the functions with which the option may be used).
In the section following we describe in detail all ANUPQ options. To continue onto the next section on-line using GAP's help browser, type:
gap> ?>
Prime := p
Specifies that the \(p\)-quotient for the prime p should be computed.
ClassBound := n
Specifies that the \(p\)-quotient to be computed has lower exponent-\(p\) class at most n. If this option is omitted a default of 63 (which is the maximum possible for the pq program) is taken, except for PqDescendants (see PqDescendants (4.4-1)) and in a special case of PqPCover (see PqPCover (4.1-3)). Let F be the argument (or start group of the process in the interactive case) for the function; then for PqDescendants the default is PClassPGroup(F) + 1, and for the special case of PqPCover the default is PClassPGroup(F).
pQuotient := Q
This option is only available for the standard presentation functions. It specifies that a \(p\)-quotient of the group argument of the function or group of the process is the pc p-group Q, where Q is of class less than the provided (or default) value of ClassBound. If pQuotient is provided, then the option Prime if also provided, is ignored; the prime p is discovered by computing PrimePGroup(Q).
Exponent := n
Specifies that the \(p\)-quotient to be computed has exponent n. For an interactive process, Exponent defaults to a previously supplied value for the process. Otherwise (and non-interactively), the default is 0, which means that no exponent law is enforced.
Relators := rels
Specifies that the relators sent to the pq program should be rels instead of the relators of the argument group F (or start group in the interactive case) of the calling function; rels should be a list of strings in the string representations of the generators of F, and F must be an fp group (even if the calling function accepts a pc group). This option provides a way of giving relators to the pq program, without having them pre-expanded by GAP, which can sometimes effect a performance loss of the order of 100 (see Section The Relators Option).
Notes
The pq program does not use / to indicate multiplication by an inverse and uses square brackets to represent (left normed) commutators. Also, even though the pq program accepts relations, all elements of rels must be in relator form, i.e. a relation of form w1 = w2 must be written as w1*(w2)^-1 and then put in a pair of double-quotes to make it a string. See the example below.
To ensure there are no syntax errors in rels, each relator is parsed for validity via PqParseWord (see PqParseWord (3.4-3)). If they are ok, a message to say so is Info-ed at InfoANUPQ level 2.
Metabelian
Specifies that the largest metabelian \(p\)-quotient subject to any other conditions specified by other options be constructed. By default this restriction is not enforced.
GroupName := name
Specifies that the pq program should refer to the group by the name name (a string). If GroupName is not set and the group has been assigned a name via SetName (see Reference: Name) it is set as the name the pq program should use. Otherwise, the "generic" name "[grp]" is set as a default.
Identities := funcs
Specifies that the pc presentation should satisfy the laws defined by each function in the list funcs. This option may be called by Pq, PqEpimorphism, or PqPCover (see Pq (4.1-1)). Each function in the list funcs must return a word in its arguments (there may be any number of arguments). Let identity be one such function in funcs. Then as each lower exponent p-class quotient is formed, instances \(\textit{identity}(\textit{w1}, \dots, \textit{wn})\) are added as relators to the pc presentation, where \(\textit{w1}, \dots, \textit{wn}\) are words in the pc generators of the quotient. At each class the class and number of pc generators is Info-ed at InfoANUPQ level 1, the number of instances is Info-ed at InfoANUPQ level 2, and the instances that are evaluated are Info-ed at InfoANUPQ level 3. As usual timing information is Info-ed at InfoANUPQ level 2; and details of the processing of each instance from the pq program (which is often quite voluminous) is Info-ed at InfoANUPQ level 3. Try the examples "B2-4-Id" and "11gp-3-Engel-Id" which demonstrate the usage of the Identities option; these are run using PqExample (see PqExample (3.4-4)). Take note of Note 1. below in relation to the example "B2-4-Id"; the companion example "B2-4" generates the same group using the Exponent option. These examples are discussed at length in Section The Identities Option and PqEvaluateIdentities Function.
Notes
Setting the InfoANUPQ level to 3 or more when setting the Identities option may slow down the computation considerably, by overloading GAP with io operations.
The Identities option is implemented at the GAP level. An identity that is just an exponent law should be specified using the Exponent option (see option Exponent), which is implemented at the C level and is highly optimised and so is much more efficient.
The number of instances of each identity tends to grow combinatorially with the class. So care should be exercised in using the Identities option, by including other restrictions, e.g. by using the ClassBound option (see option ClassBound).
OutputLevel := n
Specifies the level of "verbosity" of the information output by the ANU pq program when computing a pc presentation; n must be an integer in the range 0 to 3. OutputLevel := 0 displays at most one line of output and is the default; OutputLevel := 1 displays (usually) slightly more output and OutputLevels of 2 and 3 are two levels of verbose output. To see these messages from the pq program, the InfoANUPQ level must be set to at least 1 (see InfoANUPQ (3.3-1)). See Section Hints and Warnings regarding the use of Options for an example of how OutputLevel can be used as a troubleshooting tool.
RedoPcp
Specifies that the current pc presentation (for an interactive process) stored by the pq program be scrapped and clears the current values stored for the options Prime, ClassBound, Exponent and Metabelian and also clears the pQuotient, pQepi and pCover fields of the data record of the process.
SetupFile := filename
Non-interactively, this option directs that pq should not be called and that an input file with name filename (a string), containing the commands necessary for the ANU pq standalone, be constructed. The commands written to filename are also Info-ed behind a "ToPQ> " prompt at InfoANUPQ level 4 (see InfoANUPQ (3.3-1)). Except in the case following, the calling function returns true. If the calling function is the non-interactive version of one of Pq, PqPCover or PqEpimorphism and the group provided as argument is trivial given with an empty set of generators, then no setup file is written and fail is returned (the pq program cannot do anything useful with such a group). Interactively, SetupFile is ignored.
Note: Since commands emitted to the pq program may depend on knowing what the "current state" is, to form a setup file some "close enough guesses" may sometimes be necessary; when this occurs a warning is Info-ed at InfoANUPQ or InfoWarning level 1. To determine whether the "close enough guesses" give an accurate setup file, it is necessary to run the command without the SetupFile option, after either setting the InfoANUPQ level to at least 4 (the setup file script can then be compared with the "ToPQ> " commands that are Info-ed) or setting a pq command log file by using ToPQLog (see ToPQLog (3.4-7)).
PqWorkspace := workspace
Non-interactively, this option sets the memory used by the pq program. It sets the maximum number of integer-sized elements to allocate in its main storage array. By default, the pq program sets this figure to 10000000. Interactively, PqWorkspace is ignored; the memory used in this case may be set by giving PqStart a second argument (see PqStart (5.1-1)).
PcgsAutomorphisms
PcgsAutomorphisms := false
Let G be the group associated with the calling function (or associated interactive process). Passing the option PcgsAutomorphisms without a value (or equivalently setting it to true), specifies that a polycyclic generating sequence for the automorphism group (which must be soluble) of G, be computed and passed to the pq program. This increases the efficiency of the computation; it also prevents the pq from calling GAP for orbit-stabilizer calculations. By default, PcgsAutomorphisms is set to the value returned by IsSolvable( AutomorphismGroup( G ) ), and uses the package AutPGrp to compute AutomorphismGroup( G ) if it is installed. This flag is set to true or false in the background according to the above criterion by the function PqDescendants (see PqDescendants (4.4-1) and PqDescendants (5.3-6)).
Note: If PcgsAutomorphisms is used when the automorphism group of G is insoluble, an error message occurs.
OrderBound := n
Specifies that only descendants of size at most \(p^\textit{n}\), where n is a non-negative integer, be generated. Note that you cannot set both OrderBound and StepSize.
StepSize := n
StepSize := list
For a positive integer n, StepSize specifies that only those immediate descendants which are a factor \(p^\textit{n}\) bigger than their parent group be generated.
For a list list of positive integers such that the sum of the length of list and the exponent-\(p\) class of G is equal to the class bound defined by the option ClassBound, StepSize specifies that the integers of list are the step sizes for each additional class.
RankInitialSegmentSubgroups := n
Sets the rank of the initial segment subgroup chosen to be n. By default, this has value 0.
SpaceEfficient
Specifies that the pq program performs certain calculations of \(p\)-group generation more slowly but with greater space efficiency. This flag is frequently necessary for groups of large Frattini quotient rank. The space saving occurs because only one permutation is stored at any one time. This option is only available if the PcgsAutomorphisms flag is set to true (see option PcgsAutomorphisms). For an interactive process, SpaceEfficient defaults to a previously supplied value for the process. Otherwise (and non-interactively), SpaceEfficient is by default false.
CapableDescendants
By default, all (i.e. capable and terminal) descendants are computed. If this flag is set, only capable descendants are computed. Setting this option is equivalent to setting AllDescendants := false (see option AllDescendants), except if both CapableDescendants and AllDescendants are passed, AllDescendants is essentially ignored.
AllDescendants := false
By default, all descendants are constructed. If this flag is set to false, only capable descendants are computed. Passing AllDescendants without a value (which is equivalent to setting it to true) is superfluous. This option is provided only for backward compatibility with the GAP 3 version of the ANUPQ package, where by default AllDescendants was set to false (rather than true). It is preferable to use CapableDescendants (see option CapableDescendants).
TreeDepth := class
Specifies that the descendants tree developed by PqDescendantsTreeCoclassOne (see PqDescendantsTreeCoclassOne (A.4-1)) should be extended to class class, where class is a positive integer.
SubList := sub
Suppose that L is the list of descendants generated, then for a list sub of integers this option causes PqDescendants to return Sublist( L, sub ). If an integer n is supplied, PqDescendants returns L[n].
NumberOfSolubleAutomorphisms := n
Specifies that the number of soluble automorphisms of the automorphism group supplied by PqPGSupplyAutomorphisms (see PqPGSupplyAutomorphisms (5.9-1)) in a \(p\)-group generation calculation is n. By default, n is taken to be \(0\); n must be a non-negative integer. If \(\textit{n} \ge 0\) then a value for the option RelativeOrders (see 6.2) must also be supplied.
RelativeOrders := list
Specifies the relative orders of each soluble automorphism of the automorphism group supplied by PqPGSupplyAutomorphisms (see PqPGSupplyAutomorphisms (5.9-1)) in a \(p\)-group generation calculation. The list list must consist of n positive integers, where n is the value of the option NumberOfSolubleAutomorphisms (see 6.2). By default list is empty.
BasicAlgorithm
Specifies that an algorithm that the pq program calls its "default" algorithm be used for \(p\)-group generation. By default this algorithm is not used. If this option is supplied the settings of options RankInitialSegmentSubgroups, AllDescendants, Exponent and Metabelian are ignored.
CustomiseOutput := rec
Specifies that fine tuning of the output is desired. The record rec should have any subset (or all) of the the following fields:
perm := listwhere list is a list of booleans which determine whether the permutation group output for the automorphism group should contain: the degree, the extended automorphisms, the automorphism matrices, and the permutations, respectively.
orbit := listwhere list is a list of booleans which determine whether the orbit output of the action of the automorphism group should contain: a summary, and a complete listing of orbits, respectively. (It's possible to have both a summary and a complete listing.)
group := listwhere list is a list of booleans which determine whether the group output should contain: the standard matrix of each allowable subgroup, the presentation of reduced \(p\)-covering groups, the presentation of immediate descendants, the nuclear rank of descendants, and the \(p\)-multiplicator rank of descendants, respectively.
autgroup := listwhere list is a list of booleans which determine whether the automorphism group output should contain: the commutator matrix, the automorphism group description of descendants, and the automorphism group order of descendants, respectively.
trace := valwhere val is a boolean which if true specifies algorithm trace data is desired. By default, one does not get algorithm trace data.
Not providing a field (or mis-spelling it!), specifies that the default output is desired. As a convenience, 1 is also accepted as true, and any value that is neither 1 nor true is taken as false. Also for each list above, an unbound list entry is taken as false. Thus, for example
CustomiseOutput := rec(group := [,,1], autgroup := [,1])
specifies for the group output that only the presentation of immediate descendants is desired, for the automorphism group output only the automorphism group description of descendants should be printed, that there should be no algorithm trace data, and that the default output should be provided for the permutation group and orbit output.
StandardPresentationFile := filename
Specifies that the file to which the standard presentation is written has 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. If this option is omitted it is written to the file with the name generated by the command Filename( ANUPQData.tmpdir, "SPres" );, i.e. the file with name "SPres" in the temporary directory in which the pq program executes.
QueueFactor := n
Specifies a queue factor of n, where n must be a positive integer. This option may be used with PqNextClass (see PqNextClass (5.6-4)).
The queue factor is used when the pq program uses automorphisms to close a set of elements of the \(p\)-multiplicator under their action.
The algorithm used is a spinning algorithm: it starts with a set of vectors in echelonized form (elements of the \(p\)-multiplicator) and closes the span of these vectors under the action of the automorphisms. For this each automorphism is applied to each vector and it is checked if the result is contained in the span computed so far. If not, the span becomes bigger and the vector is put into a queue and the automorphisms are applied to this vector at a later stage. The process terminates when the automorphisms have been applied to all vectors and no new vectors have been produced.
For each new vector it is decided, if its processing should be delayed. If the vector contains too many non-zero entries, it is put into a second queue. The elements in this queue are processed only when there are no elements in the first queue left.
The queue factor is a percentage figure. A vector is put into the second queue if the percentage of its non-zero entries exceeds the queue factor.
Bounds := list
Specifies a lower and upper bound on the indices of a list, where list is a pair of positive non-decreasing integers. See PqDisplayStructure (5.7-23) and PqDisplayAutomorphisms (5.7-24) where this option may be used.
PrintAutomorphisms := list
Specifies that automorphism matrices be printed.
PrintPermutations := list
Specifies that permutations of the subgroups be printed.
Filename := string
Specifies that an output or input file to be written to or read from by the pq program should have the name string.
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