6 New **GAP** Objects and Utility Functions provided by the
**AtlasRep** Package

6.1 Straight Line Decisions

6.1-1 IsStraightLineDecision

6.1-2 LinesOfStraightLineDecision

6.1-3 NrInputsOfStraightLineDecision

6.1-4 ScanStraightLineDecision

6.1-5 StraightLineDecision

6.1-6 ResultOfStraightLineDecision

6.1-7 Semi-Presentations and Presentations

6.1-8 AsStraightLineDecision

6.1-9 StraightLineProgramFromStraightLineDecision

6.1-1 IsStraightLineDecision

6.1-2 LinesOfStraightLineDecision

6.1-3 NrInputsOfStraightLineDecision

6.1-4 ScanStraightLineDecision

6.1-5 StraightLineDecision

6.1-6 ResultOfStraightLineDecision

6.1-7 Semi-Presentations and Presentations

6.1-8 AsStraightLineDecision

6.1-9 StraightLineProgramFromStraightLineDecision

This chapter describes **GAP** objects and functions that are provided by the **AtlasRep** package but that might be of general interest.

The new objects are straight line decisions (see Section 6.1) and black box programs (see Section 6.2).

The new functions are concerned with representations of minimal degree, see Section 6.3, and a JSON interface, see Section 6.4.

*Straight line decisions* are similar to straight line programs (see Section Reference: Straight Line Programs) but return `true`

or `false`

. A straight line decision checks whether its inputs have some property. An important example is to check whether a given list of group generators is in fact a list of standard generators (cf. Section3.3) for this group.

A straight line decision in **GAP** is represented by an object in the filter `IsStraightLineDecision`

(6.1-1) that stores a list of "lines" each of which has one of the following three forms.

a nonempty dense list \(l\) of integers,

a pair \([ l, i ]\) where \(l\) is a list of form 1. and \(i\) is a positive integer,

a list \([ \)

`"Order"`

\(, i, n ]\) where \(i\) and \(n\) are positive integers.

The first two forms have the same meaning as for straight line programs (see Section Reference: Straight Line Programs), the last form means a check whether the element stored at the \(i\)-th label has the order \(n\).

For the meaning of the list of lines, see `ResultOfStraightLineDecision`

(6.1-6).

Straight line decisions can be constructed using `StraightLineDecision`

(6.1-5), defining attributes for straight line decisions are `NrInputsOfStraightLineDecision`

(6.1-3) and `LinesOfStraightLineDecision`

(6.1-2), an operation for straight line decisions is `ResultOfStraightLineDecision`

(6.1-6).

Special methods applicable to straight line decisions are installed for the operations `Display`

(Reference: Display), `IsInternallyConsistent`

(Reference: IsInternallyConsistent), `PrintObj`

(Reference: PrintObj), and `ViewObj`

(Reference: ViewObj).

For a straight line decision `prog`, the default `Display`

(Reference: Display) method prints the interpretation of `prog` as a sequence of assignments of associative words and of order checks; a record with components `gensnames`

(with value a list of strings) and `listname`

(a string) may be entered as second argument of `Display`

(Reference: Display), in this case these names are used, the default for `gensnames`

is `[ g1, g2, `

\(\ldots\)` ]`

, the default for `listname` is \(r\).

`‣ IsStraightLineDecision` ( obj ) | ( category ) |

Each straight line decision in **GAP** lies in the filter `IsStraightLineDecision`

.

`‣ LinesOfStraightLineDecision` ( prog ) | ( operation ) |

Returns: the list of lines that define the straight line decision.

This defining attribute for the straight line decision `prog` (see `IsStraightLineDecision`

(6.1-1)) corresponds to `LinesOfStraightLineProgram`

(Reference: LinesOfStraightLineProgram) for straight line programs.

gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ], > [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] ); <straight line decision> gap> LinesOfStraightLineDecision( dec ); [ [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ]

`‣ NrInputsOfStraightLineDecision` ( prog ) | ( operation ) |

Returns: the number of inputs required for the straight line decision.

This defining attribute corresponds to `NrInputsOfStraightLineProgram`

(Reference: NrInputsOfStraightLineProgram).

gap> NrInputsOfStraightLineDecision( dec ); 2

`‣ ScanStraightLineDecision` ( string ) | ( function ) |

Returns: a record containing the straight line decision, or `fail`

.

Let `string` be a string that encodes a straight line decision in the sense that it consists of the lines listed for `ScanStraightLineProgram`

(7.4-1), except that `oup`

lines are not allowed, and instead lines of the following form may occur.

`chor`

\(a\) \(b\)means that it is checked whether the order of the element at label \(a\) is \(b\).

`ScanStraightLineDecision`

returns a record containing as the value of its component `program`

the corresponding **GAP** straight line decision (see `IsStraightLineDecision`

(6.1-1)) if the input string satisfies the syntax rules stated above, and returns `fail`

otherwise. In the latter case, information about the first corrupted line of the program is printed if the info level of `InfoCMeatAxe`

(7.1-2) is at least \(1\).

gap> str:= "inp 2\nchor 1 2\nchor 2 3\nmu 1 2 3\nchor 3 5";; gap> prg:= ScanStraightLineDecision( str ); rec( program := <straight line decision> ) gap> prg:= prg.program;; gap> Display( prg ); # input: r:= [ g1, g2 ]; # program: if Order( r[1] ) <> 2 then return false; fi; if Order( r[2] ) <> 3 then return false; fi; r[3]:= r[1]*r[2]; if Order( r[3] ) <> 5 then return false; fi; # return value: true

`‣ StraightLineDecision` ( lines[, nrgens] ) | ( function ) |

`‣ StraightLineDecisionNC` ( lines[, nrgens] ) | ( function ) |

Returns: the straight line decision given by the list of lines.

Let `lines` be a list of lists that defines a unique straight line decision (see `IsStraightLineDecision`

(6.1-1)); in this case `StraightLineDecision`

returns this program, otherwise an error is signalled. The optional argument `nrgens` specifies the number of input generators of the program; if a list of integers (a line of form 1. in the definition above) occurs in `lines` then this number is not determined by `lines` and therefore *must* be specified by the argument `nrgens`; if not then `StraightLineDecision`

returns `fail`

.

`StraightLineDecisionNC`

does the same as `StraightLineDecision`

, except that the internal consistency of the program is not checked.

`‣ ResultOfStraightLineDecision` ( prog, gens[, orderfunc] ) | ( operation ) |

Returns: `true`

if all checks succeed, otherwise `false`

.

`ResultOfStraightLineDecision`

evaluates the straight line decision (see `IsStraightLineDecision`

(6.1-1)) `prog` at the group elements in the list `gens`.

The function for computing the order of a group element can be given as the optional argument `orderfunc`. For example, this may be a function that gives up at a certain limit if one has to be aware of extremely huge orders in failure cases.

The *result* of a straight line decision with lines \(p_1, p_2, \ldots, p_k\) when applied to `gens` is defined as follows.

**(a)**First a list \(r\) of intermediate values is initialized with a shallow copy of

`gens`.**(b)**For \(i \leq k\), before the \(i\)-th step, let \(r\) be of length \(n\). If \(p_i\) is the external representation of an associative word in the first \(n\) generators then the image of this word under the homomorphism that is given by mapping \(r\) to these first \(n\) generators is added to \(r\). If \(p_i\) is a pair \([ l, j ]\), for a list \(l\), then the same element is computed, but instead of being added to \(r\), it replaces the \(j\)-th entry of \(r\). If \(p_i\) is a triple \([ \)

`"Order"`

\(, i, n ]\) then it is checked whether the order of \(r[i]\) is \(n\); if not then`false`

is returned immediately.**(c)**If all \(k\) lines have been processed and no order check has failed then

`true`

is returned.

Here are some examples.

gap> dec:= StraightLineDecision( [ ], 1 ); <straight line decision> gap> ResultOfStraightLineDecision( dec, [ () ] ); true

The above straight line decision `dec`

returns `true`

–for *any* input of the right length.

gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ], > [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] ); <straight line decision> gap> LinesOfStraightLineDecision( dec ); [ [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] gap> ResultOfStraightLineDecision( dec, [ (), () ] ); false gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,4,5) ] ); true

The above straight line decision admits two inputs; it tests whether the orders of the inputs are \(2\) and \(3\), and the order of their product is \(5\).

We can associate a *finitely presented group* \(F / R\) to each straight line decision `dec`, say, as follows. The free generators of the free group \(F\) are in bijection with the inputs, and the defining relators generating \(R\) as a normal subgroup of \(F\) are given by those words \(w^k\) for which `dec` contains a check whether the order of \(w\) equals \(k\).

So if `dec` returns `true`

for the input list \([ g_1, g_2, \ldots, g_n ]\) then mapping the free generators of \(F\) to the inputs defines an epimorphism \(\Phi\) from \(F\) to the group \(G\), say, that is generated by these inputs, such that \(R\) is contained in the kernel of \(\Phi\).

(Note that "satisfying `dec`" is a stronger property than "satisfying a presentation". For example, \(\langle x \mid x^2 = x^3 = 1 \rangle\) is a presentation for the trivial group, but the straight line decision that checks whether the order of \(x\) is both \(2\) and \(3\) clearly always returns `false`

.)

**AtlasRep** supports the following two kinds of straight line decisions.

A

*presentation*is a straight line decision`dec`that is defined for a set of standard generators of a group \(G\) and that returns`true`

if and only if the list of inputs is in fact a sequence of such standard generators for \(G\). In other words, the relators derived from the order checks in the way described above are defining relators for \(G\), and moreover these relators are words in terms of standard generators. (In particular the kernel of the map \(\Phi\) equals \(R\) whenever`dec`returns`true`

.)A

*semi-presentation*is a straight line decision`dec`that is defined for a set of standard generators of a group \(G\) and that returns`true`

for a list of inputs*that is known to generate a group isomorphic with \(G\)*if and only if these inputs form in fact a sequence of standard generators for \(G\). In other words, the relators derived from the order checks in the way described above are*not necessarily defining relators*for \(G\), but if we assume that the \(g_i\) generate \(G\) then they are standard generators. (In particular, \(F / R\) may be a larger group than \(G\) but in this case \(\Phi\) maps the free generators of \(F\) to standard generators of \(G\).)More about semi-presentations can be found in [NW05].

Available presentations and semi-presentations are listed by `DisplayAtlasInfo`

(3.5-1), they can be accessed via `AtlasProgram`

(3.5-4). (Clearly each presentation is also a semi-presentation. So a semi-presentation for some standard generators of a group is regarded as available whenever a presentation for these standard generators and this group is available.)

Note that different groups can have the same semi-presentation. We illustrate this with an example that is mentioned in [NW05]. The groups \(L_2(7) \cong L_3(2)\) and \(L_2(8)\) are generated by elements of the orders \(2\) and \(3\) such that their product has order \(7\), and no further conditions are necessary to define standard generators.

gap> check:= AtlasProgram( "L2(8)", "check" ); rec( groupname := "L2(8)", identifier := [ "L2(8)", "L28G1-check1", 1, 1 ], program := <straight line decision>, standardization := 1, version := "1" ) gap> gens:= AtlasGenerators( "L2(8)", 1 ); rec( charactername := "1a+8a", constituents := [ 1, 6 ], contents := "core", generators := [ (1,2)(3,4)(6,7)(8,9), (1,3,2)(4,5,6)(7,8,9) ], groupname := "L2(8)", id := "", identifier := [ "L2(8)", [ "L28G1-p9B0.m1", "L28G1-p9B0.m2" ], 1, 9 ], isPrimitive := true, maxnr := 1, p := 9, rankAction := 2, repname := "L28G1-p9B0", repnr := 1, size := 504, stabilizer := "2^3:7", standardization := 1, transitivity := 3, type := "perm" ) gap> ResultOfStraightLineDecision( check.program, gens.generators ); true gap> gens:= AtlasGenerators( "L3(2)", 1 ); rec( contents := "core", generators := [ (2,4)(3,5), (1,2,3)(5,6,7) ], groupname := "L3(2)", id := "a", identifier := [ "L3(2)", [ "L27G1-p7aB0.m1", "L27G1-p7aB0.m2" ], 1, 7 ], isPrimitive := true, maxnr := 1, p := 7, rankAction := 2, repname := "L27G1-p7aB0", repnr := 1, size := 168, stabilizer := "S4", standardization := 1, transitivity := 2, type := "perm" ) gap> ResultOfStraightLineDecision( check.program, gens.generators ); true

`‣ AsStraightLineDecision` ( bbox ) | ( attribute ) |

Returns: an equivalent straight line decision for the given black box program, or `fail`

.

For a black box program (see `IsBBoxProgram`

(6.2-1)) `bbox`, `AsStraightLineDecision`

returns a straight line decision (see `IsStraightLineDecision`

(6.1-1)) with the same output as `bbox`, in the sense of `AsBBoxProgram`

(6.2-5), if such a straight line decision exists, and `fail`

otherwise.

gap> lines:= [ [ "Order", 1, 2 ], [ "Order", 2, 3 ], > [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ];; gap> dec:= StraightLineDecision( lines, 2 ); <straight line decision> gap> bboxdec:= AsBBoxProgram( dec ); <black box program> gap> asdec:= AsStraightLineDecision( bboxdec ); <straight line decision> gap> LinesOfStraightLineDecision( asdec ); [ [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ]

`‣ StraightLineProgramFromStraightLineDecision` ( dec ) | ( operation ) |

Returns: the straight line program associated to the given straight line decision.

For a straight line decision `dec` (see `IsStraightLineDecision`

(6.1-1), `StraightLineProgramFromStraightLineDecision`

returns the straight line program (see `IsStraightLineProgram`

(Reference: IsStraightLineProgram) obtained by replacing each line of type 3. (i.e, each order check) by an assignment of the power in question to a new slot, and by declaring the list of these elements as the return value.

This means that the return value describes exactly the defining relators of the presentation that is associated to the straight line decision, see 6.1-7.

For example, one can use the return value for printing the relators with `StringOfResultOfStraightLineProgram`

(Reference: StringOfResultOfStraightLineProgram), or for explicitly constructing the relators as words in terms of free generators, by applying `ResultOfStraightLineProgram`

(Reference: ResultOfStraightLineProgram) to the program and to these generators.

gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ], > [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] ); <straight line decision> gap> prog:= StraightLineProgramFromStraightLineDecision( dec ); <straight line program> gap> Display( prog ); # input: r:= [ g1, g2 ]; # program: r[3]:= r[1]*r[2]; r[4]:= r[1]^2; r[5]:= r[2]^3; r[6]:= r[3]^5; # return values: [ r[4], r[5], r[6] ] gap> StringOfResultOfStraightLineProgram( prog, [ "a", "b" ] ); "[ a^2, b^3, (ab)^5 ]" gap> gens:= GeneratorsOfGroup( FreeGroup( "a", "b" ) ); [ a, b ] gap> ResultOfStraightLineProgram( prog, gens ); [ a^2, b^3, (a*b)^5 ]

*Black box programs* formalize the idea that one takes some group elements, forms arithmetic expressions in terms of them, tests properties of these expressions, executes conditional statements (including jumps inside the program) depending on the results of these tests, and eventually returns some result.

A specification of the language can be found in [Nic06], see also

http://brauer.maths.qmul.ac.uk/Atlas/info/blackbox.html.

The *inputs* of a black box program may be explicit group elements, and the program may also ask for random elements from a given group. The *program steps* form products, inverses, conjugates, commutators, etc. of known elements, *tests* concern essentially the orders of elements, and the *result* is a list of group elements or `true`

or `false`

or `fail`

.

Examples that can be modeled by black box programs are

*straight line programs*,which require a fixed number of input elements and form arithmetic expressions of elements but do not use random elements, tests, conditional statements and jumps; the return value is always a list of elements; these programs are described in Section Reference: Straight Line Programs.

*straight line decisions*,which differ from straight line programs only in the sense that also order tests are admissible, and that the return value is

`true`

if all these tests are satisfied, and`false`

as soon as the first such test fails; they are described in Section 6.1.*scripts for finding standard generators*,which take a group and a function to generate a random element in this group but no explicit input elements, admit all control structures, and return either a list of standard generators or

`fail`

; see`ResultOfBBoxProgram`

(6.2-4) for examples.

In the case of general black box programs, currently **GAP** provides only the possibility to read an existing program via `ScanBBoxProgram`

(6.2-2), and to run the program using `RunBBoxProgram`

(6.2-3). It is not our aim to write such programs in **GAP**.

The special case of the "find" scripts mentioned above is also admissible as an argument of `ResultOfBBoxProgram`

(6.2-4), which returns either the set of found generators or `fail`

.

Contrary to the general situation, more support is provided for straight line programs and straight line decisions in **GAP**, see Section Reference: Straight Line Programs for functions that manipulate them (compose, restrict etc.).

The functions `AsStraightLineProgram`

(6.2-6) and `AsStraightLineDecision`

(6.1-8) can be used to transform a general black box program object into a straight line program or a straight line decision if this is possible.

Conversely, one can create an equivalent general black box program from a straight line program or from a straight line decision with `AsBBoxProgram`

(6.2-5).

Computing a straight line program related to a given straight line decision is supported in the sense of `StraightLineProgramFromStraightLineDecision`

(6.1-9).

Note that none of these three kinds of objects is a special case of another: Running a black box program with `RunBBoxProgram`

(6.2-3) yields a record, running a straight line program with `ResultOfStraightLineProgram`

(Reference: ResultOfStraightLineProgram) yields a list of elements, and running a straight line decision with `ResultOfStraightLineDecision`

(6.1-6) yields `true`

or `false`

.

`‣ IsBBoxProgram` ( obj ) | ( category ) |

Each black box program in **GAP** lies in the filter `IsBBoxProgram`

.

`‣ ScanBBoxProgram` ( string ) | ( function ) |

Returns: a record containing the black box program encoded by the input string, or `fail`

.

For a string `string` that describes a black box program, e.g., the return value of `StringFile`

(GAPDoc: StringFile), `ScanBBoxProgram`

computes this black box program. If this is successful then the return value is a record containing as the value of its component `program`

the corresponding **GAP** object that represents the program, otherwise `fail`

is returned.

As the first example, we construct a black box program that tries to find standard generators for the alternating group \(A_5\); these standard generators are any pair of elements of the orders \(2\) and \(3\), respectively, such that their product has order \(5\).

gap> findstr:= "\ > set V 0\n\ > lbl START1\n\ > rand 1\n\ > ord 1 A\n\ > incr V\n\ > if V gt 100 then timeout\n\ > if A notin 1 2 3 5 then fail\n\ > if A noteq 2 then jmp START1\n\ > lbl START2\n\ > rand 2\n\ > ord 2 B\n\ > incr V\n\ > if V gt 100 then timeout\n\ > if B notin 1 2 3 5 then fail\n\ > if B noteq 3 then jmp START2\n\ > # The elements 1 and 2 have the orders 2 and 3, respectively.\n\ > set X 0\n\ > lbl CONJ\n\ > incr X\n\ > if X gt 100 then timeout\n\ > rand 3\n\ > cjr 2 3\n\ > mu 1 2 4 # ab\n\ > ord 4 C\n\ > if C notin 2 3 5 then fail\n\ > if C noteq 5 then jmp CONJ\n\ > oup 2 1 2";; gap> find:= ScanBBoxProgram( findstr ); rec( program := <black box program> )

The second example is a black box program that checks whether its two inputs are standard generators for \(A_5\).

gap> checkstr:= "\ > chor 1 2\n\ > chor 2 3\n\ > mu 1 2 3\n\ > chor 3 5";; gap> check:= ScanBBoxProgram( checkstr ); rec( program := <black box program> )

`‣ RunBBoxProgram` ( prog, G, input, options ) | ( function ) |

Returns: a record describing the result and the statistics of running the black box program `prog`, or `fail`

, or the string `"timeout"`

.

For a black box program `prog`, a group `G`, a list `input` of group elements, and a record `options`, `RunBBoxProgram`

applies `prog` to `input`, where `G` is used only to compute random elements.

The return value is `fail`

if a syntax error or an explicit `fail`

statement is reached at runtime, and the string `"timeout"`

if a `timeout`

statement is reached. (The latter might mean that the random choices were unlucky.) Otherwise a record with the following components is returned.

`gens`

a list of group elements, bound if an

`oup`

statement was reached,`result`

`true`

if a`true`

statement was reached,`false`

if either a`false`

statement or a failed order check was reached,

The other components serve as statistical information about the numbers of the various operations (`multiply`

, `invert`

, `power`

, `order`

, `random`

, `conjugate`

, `conjugateinplace`

, `commutator`

), and the runtime in milliseconds (`timetaken`

).

The following components of `options` are supported.

`randomfunction`

the function called with argument

`G`in order to compute a random element of`G`(default`PseudoRandom`

(Reference: PseudoRandom))`orderfunction`

the function for computing element orders (default

`Order`

(Reference: Order)),`quiet`

if

`true`

then ignore`echo`

statements (default`false`

),`verbose`

if

`true`

then print information about the line that is currently processed, and about order checks (default`false`

),`allowbreaks`

if

`true`

then call`Error`

(Reference: Error) when a`break`

statement is reached, otherwise ignore`break`

statements (default`true`

).

As an example, we run the black box programs constructed in the example for `ScanBBoxProgram`

(6.2-2).

gap> g:= AlternatingGroup( 5 );; gap> res:= RunBBoxProgram( find.program, g, [], rec() );; gap> IsBound( res.gens ); IsBound( res.result ); true false gap> List( res.gens, Order ); [ 2, 3 ] gap> Order( Product( res.gens ) ); 5 gap> res:= RunBBoxProgram( check.program, "dummy", res.gens, rec() );; gap> IsBound( res.gens ); IsBound( res.result ); false true gap> res.result; true gap> othergens:= GeneratorsOfGroup( g );; gap> res:= RunBBoxProgram( check.program, "dummy", othergens, rec() );; gap> res.result; false

`‣ ResultOfBBoxProgram` ( prog, G ) | ( function ) |

Returns: a list of group elements or `true`

, `false`

, `fail`

, or the string `"timeout"`

.

This function calls `RunBBoxProgram`

(6.2-3) with the black box program `prog` and second argument either a group or a list of group elements; the default options of `RunBBoxProgram`

(6.2-3) are assumed. The return value is `fail`

if this call yields `fail`

, otherwise the `gens`

component of the result, if bound, or the `result`

component if not.

Note that a *group* `G` is used as the second argument in the call of `RunBBoxProgram`

(6.2-3) (the source for random elements), whereas a *list* `G` is used as the third argument (the inputs).

As an example, we run the black box programs constructed in the example for `ScanBBoxProgram`

(6.2-2).

gap> g:= AlternatingGroup( 5 );; gap> res:= ResultOfBBoxProgram( find.program, g );; gap> List( res, Order ); [ 2, 3 ] gap> Order( Product( res ) ); 5 gap> res:= ResultOfBBoxProgram( check.program, res ); true gap> othergens:= GeneratorsOfGroup( g );; gap> res:= ResultOfBBoxProgram( check.program, othergens ); false

`‣ AsBBoxProgram` ( slp ) | ( attribute ) |

Returns: an equivalent black box program for the given straight line program or straight line decision.

Let `slp` be a straight line program (see `IsStraightLineProgram`

(Reference: IsStraightLineProgram)) or a straight line decision (see `IsStraightLineDecision`

(6.1-1)). Then `AsBBoxProgram`

returns a black box program `bbox` (see `IsBBoxProgram`

(6.2-1)) with the "same" output as `slp`, in the sense that `ResultOfBBoxProgram`

(6.2-4) yields the same result for `bbox` as `ResultOfStraightLineProgram`

(Reference: ResultOfStraightLineProgram) or `ResultOfStraightLineDecision`

(6.1-6), respectively, for `slp`.

gap> f:= FreeGroup( "x", "y" );; gens:= GeneratorsOfGroup( f );; gap> slp:= StraightLineProgram( [ [1,2,2,3], [3,-1] ], 2 ); <straight line program> gap> ResultOfStraightLineProgram( slp, gens ); y^-3*x^-2 gap> bboxslp:= AsBBoxProgram( slp ); <black box program> gap> ResultOfBBoxProgram( bboxslp, gens ); [ y^-3*x^-2 ] gap> lines:= [ [ "Order", 1, 2 ], [ "Order", 2, 3 ], > [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ];; gap> dec:= StraightLineDecision( lines, 2 ); <straight line decision> gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,3,5) ] ); true gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,3,4) ] ); false gap> bboxdec:= AsBBoxProgram( dec ); <black box program> gap> ResultOfBBoxProgram( bboxdec, [ (1,2)(3,4), (1,3,5) ] ); true gap> ResultOfBBoxProgram( bboxdec, [ (1,2)(3,4), (1,3,4) ] ); false

`‣ AsStraightLineProgram` ( bbox ) | ( attribute ) |

Returns: an equivalent straight line program for the given black box program, or `fail`

.

For a black box program (see `AsBBoxProgram`

(6.2-5)) `bbox`, `AsStraightLineProgram`

returns a straight line program (see `IsStraightLineProgram`

(Reference: IsStraightLineProgram)) with the same output as `bbox` if such a straight line program exists, and `fail`

otherwise.

gap> Display( AsStraightLineProgram( bboxslp ) ); # input: r:= [ g1, g2 ]; # program: r[3]:= r[1]^2; r[4]:= r[2]^3; r[5]:= r[3]*r[4]; r[3]:= r[5]^-1; # return values: [ r[3] ] gap> AsStraightLineProgram( bboxdec ); fail

This section deals with minimal degrees of permutation and matrix representations. We do not provide an algorithm that computes these degrees for an arbitrary group, we only provide some tools for evaluating known databases, mainly concerning "bicyclic extensions" (see [CCN+85, Section 6.5]) of simple groups, in order to derive the minimal degrees, see Section 6.3-4.

In the **AtlasRep** package, this information can be used for prescribing "minimality conditions" in `DisplayAtlasInfo`

(3.5-1), `OneAtlasGeneratingSetInfo`

(3.5-6), and `AllAtlasGeneratingSetInfos`

(3.5-7). An overview of the stored minimal degrees can be shown with `BrowseMinimalDegrees`

(3.6-1).

`‣ MinimalRepresentationInfo` ( grpname, conditions ) | ( function ) |

Returns: a record with the components `value`

and `source`

, or `fail`

Let `grpname` be the **GAP** name of a group \(G\), say. If the information described by `conditions` about minimal representations of this group can be computed or is stored then `MinimalRepresentationInfo`

returns a record with the components `value`

and `source`

, otherwise `fail`

is returned.

The following values for `conditions` are supported.

If

`conditions`is`NrMovedPoints`

(Reference: NrMovedPoints for a permutation) then`value`

, if known, is the degree of a minimal faithful (not necessarily transitive) permutation representation for \(G\).If

`conditions`consists of`Characteristic`

(Reference: Characteristic) and a prime integer`p`then`value`

, if known, is the dimension of a minimal faithful (not necessarily irreducible) matrix representation in characteristic`p`for \(G\).If

`conditions`consists of`Size`

(Reference: Size) and a prime power`q`then`value`

, if known, is the dimension of a minimal faithful (not necessarily irreducible) matrix representation over the field of size`q`for \(G\).

In all cases, the value of the component `source`

is a list of strings that describe sources of the information, which can be the ordinary or modular character table of \(G\) (see [CCN+85], [JLPW95], [HL89]), the table of marks of \(G\), or [Jan05]. For an overview of minimal degrees of faithful matrix representations for sporadic simple groups and their covering groups, see also

http://www.math.rwth-aachen.de/~MOC/mindeg/.

Note that `MinimalRepresentationInfo`

cannot provide any information about minimal representations over prescribed fields in characteristic zero.

Information about groups that occur in the **AtlasRep** package is precomputed in `MinimalRepresentationInfoData`

(6.3-2), so the packages **CTblLib** and **TomLib** are not needed when `MinimalRepresentationInfo`

is called for these groups. (The only case that is not covered by this list is that one asks for the minimal degree of matrix representations over a prescribed field in characteristic coprime to the group order.)

One of the following strings can be given as an additional last argument.

`"cache"`

means that the function tries to compute (and then store) values that are not stored in

`MinimalRepresentationInfoData`

(6.3-2), but stored values are preferred; this is also the default.`"lookup"`

means that stored values are returned but the function does not attempt to compute values that are not stored in

`MinimalRepresentationInfoData`

(6.3-2).`"recompute"`

means that the function always tries to compute the desired value, and checks the result against stored values.

gap> MinimalRepresentationInfo( "A5", NrMovedPoints ); rec( source := [ "computed (alternating group)", "computed (char. table)", "computed (subgroup tables)", "computed (subgroup tables, known repres.)", "computed (table of marks)" ], value := 5 ) gap> MinimalRepresentationInfo( "A5", Characteristic, 2 ); rec( source := [ "computed (char. table)" ], value := 2 ) gap> MinimalRepresentationInfo( "A5", Size, 2 ); rec( source := [ "computed (char. table)" ], value := 4 )

`‣ MinimalRepresentationInfoData` | ( global variable ) |

This is a record whose components are **GAP** names of groups for which information about minimal permutation and matrix representations were known in advance or have been computed in the current **GAP** session. The value for the group \(G\), say, is a record with the following components.

`NrMovedPoints`

a record with the components

`value`

(the degree of a smallest faithful permutation representation of \(G\)) and`source`

(a string describing the source of this information).`Characteristic`

a record whose components are at most

`0`

and strings corresponding to prime integers, each bound to a record with the components`value`

(the degree of a smallest faithful matrix representation of \(G\) in this characteristic) and`source`

(a string describing the source of this information).`CharacteristicAndSize`

a record whose components are strings corresponding to prime integers

`p`, each bound to a record with the components`sizes`

(a list of powers`q`of`p`),`dimensions`

(the corresponding list of minimal dimensions of faithful matrix representations of \(G\) over a field of size`q`),`sources`

(the corresponding list of strings describing the source of this information), and`complete`

(a record with the components`val`

(`true`

if the minimal dimension over*any*finite field in characteristic`p`can be derived from the values in the record, and`false`

otherwise) and`source`

(a string describing the source of this information)).

The values are set by `SetMinimalRepresentationInfo`

(6.3-3).

`‣ SetMinimalRepresentationInfo` ( grpname, op, value, source ) | ( function ) |

Returns: `true`

if the values were successfully set, `false`

if stored values contradict the given ones.

This function sets an entry in `MinimalRepresentationInfoData`

(6.3-2) for the group \(G\), say, with **GAP** name `grpname`.

Supported values for `op` are

`"NrMovedPoints"`

(see`NrMovedPoints`

(Reference: NrMovedPoints for a permutation)), which means that`value`is the degree of minimal faithful (not necessarily transitive) permutation representations of \(G\),a list of length two with first entry

`"Characteristic"`

(see`Characteristic`

(Reference: Characteristic)) and second entry`char`either zero or a prime integer, which means that`value`is the dimension of minimal faithful (not necessarily irreducible) matrix representations of \(G\) in characteristic`char`,a list of length two with first entry

`"Size"`

(see`Size`

(Reference: Size)) and second entry a prime power`q`, which means that`value`is the dimension of minimal faithful (not necessarily irreducible) matrix representations of \(G\) over the field with`q`elements, anda list of length three with first entry

`"Characteristic"`

(see`Characteristic`

(Reference: Characteristic)), second entry a prime integer`p`, and third entry the string`"complete"`

, which means that the information stored for characteristic`p`is complete in the sense that for any given power \(q\) of`p`, the minimal faithful degree over the field with \(q\) elements equals that for the largest stored field size of which \(q\) is a power.

In each case, `source` is a string describing the source of the data; *computed* values are detected from the prefix `"comp"`

of `source`.

If the intended value is already stored and differs from `value` then an error message is printed.

gap> SetMinimalRepresentationInfo( "A5", "NrMovedPoints", 5, > "computed (alternating group)" ); true gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 0 ], 3, > "computed (char. table)" ); true gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 2 ], 2, > "computed (char. table)" ); true gap> SetMinimalRepresentationInfo( "A5", [ "Size", 2 ], 4, > "computed (char. table)" ); true gap> SetMinimalRepresentationInfo( "A5", [ "Size", 4 ], 2, > "computed (char. table)" ); true gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 3 ], 3, > "computed (char. table)" ); true

The information about the minimal degree of a faithful *matrix representation* of \(G\) in a given characteristic or over a given field in positive characteristic is derived from the relevant (ordinary or modular) character table of \(G\), except in a few cases where this table itself is not known but enough information about the degrees is available in [HL89] and [Jan05].

The following criteria are used for deriving the minimal degree of a faithful *permutation representation* of \(G\) from the information in the **GAP** libraries of character tables and of tables of marks.

If the name of \(G\) has the form

`"A`

\(n\)`"`

or`"A`

\(n\)`.2"`

(denoting alternating and symmetric groups, respectively) then the minimal degree is \(n\), except if \(n\) is smaller than \(3\) or \(2\), respectively.If the name of \(G\) has the form

`"L2(`

\(q\)`)"`

(denoting projective special linear groups in dimension two) then the minimal degree is \(q + 1\), except if \(q \in \{ 2, 3, 5, 7, 9, 11 \}\), see [Hup67, Satz II.8.28].If the largest maximal subgroup of \(G\) is core-free then the index of this subgroup is the minimal degree. (This is used when the two character tables in question and the class fusion are available in

**GAP**'s Character Table Library [Bre13]; this happens for many character tables of simple groups.)If \(G\) has a unique minimal normal subgroup then each minimal faithful permutation representation is transitive. (Note that the core of each point stabilizer is either trivial or contains the unique minimal normal subgroup.)

In this case, the minimal degree can be computed directly from the information in the table of marks of \(G\) if this is available in

**GAP**'s Library of Tables of Marks [NMP18].Suppose that the largest maximal subgroup of \(G\) is not core-free but simple and normal in \(G\), and that the other maximal subgroups of \(G\) are core-free. In this case, we take the minimum of the indices of the core-free maximal subgroups and of the product of index and minimal degree of the normal maximal subgroup. (This suffices since no core-free subgroup of the whole group can contain a nontrivial normal subgroup of a normal maximal subgroup.)

Let \(N\) be the unique minimal normal subgroup of \(G\), and assume that \(G/N\) is simple and has minimal degree \(n\), say. If there is a subgroup \(U\) of index \(n \cdot |N|\) in \(G\) that intersects \(N\) trivially then the minimal degree of \(G\) is \(n \cdot |N|\). (This is used for the case that \(N\) is central in \(G\) and \(N \times U\) occurs as a subgroup of \(G\).)

If we know a subgroup of \(G\) whose minimal degree is \(n\), say, and if we know either (a class fusion from) a core-free subgroup of index \(n\) in \(G\) or a faithful permutation representation of degree \(n\) for \(G\) then \(n\) is the minimal degree for \(G\). (This happens often for tables of almost simple groups.)

We define a mapping between certain **GAP** objects and JSON (JavaScript Object Notation) texts (see [JSO14]), as follows.

The three

**GAP**values`true`

,`false`

, and`fail`

correspond to the JSON texts`true`

,`false`

, and`null`

, respectively.**GAP**strings correspond to JSON strings; special characters in a**GAP**string (control characters ASCII \(0\) to \(31\), backslash and double quote) are mapped as defined in JSON's specification, and other ASCII characters are kept as they are; if a**GAP**string contains non-ASCII characters, it is assumed that it is UTF-8 encoded, and one may choose either to keep non-ASCII characters as they are, or to create an ASCII only JSON string, using JSON's syntax for Unicode code points ("`\uXXXX`

"); in the other direction, JSON strings are assumed to be UTF-8 encoded, and are mapped to UTF-8 encoded**GAP**strings, by keeping the non-ASCII characters and converting substrings of the form`\uXXXX`

accordingly.**GAP**integers (in the sense of`IsInt`

(Reference: IsInt)) are mapped to JSON numbers that consist of digits and optionally a leading sign character`-`

; in the other direction, JSON numbers of this form and also JSON numbers that involve no decimal dots and have no negative exponent (for example`"2e3"`

) are mapped to**GAP**integers.**GAP**rationals (in the sense of`IsRat`

(Reference: IsRat)) which are not integers are represented by JSON floating point numbers; the JSON representation (and hence the precision) is given by first applying`Float`

(Reference: Float) and then`String`

(Reference: String).**GAP**floats (in the sense of Chapter Reference: Floats in the**GAP**Reference Manual) are mapped to JSON floating point numbers; the JSON representation (and hence the precision) is given by applying`String`

(Reference: String); in the other direction, JSON numbers that involve a decimal dot or a negative exponent are mapped to**GAP**floats.(Nested and not self-referential) dense

**GAP**lists of objects correspond to JSON arrays such that the list entries correspond to each other. (Note that JSON does not support non-dense arrays.)(Nested and not self-referential)

**GAP**records correspond to JSON objects such that both labels (which are strings in**GAP**and JSON) and values correspond to each other.

The **GAP** functions `AGR.JsonText`

(6.4-2) and `AGR.GapObjectOfJsonText`

(6.4-3) can be used to create a JSON text from a suitable **GAP** object and the **GAP** object that corresponds to a given JSON text, respectively.

Note that the composition of the two functions is in general *not* the identity mapping, because `AGR.JsonText`

(6.4-2) accepts non-integer rationals, whereas `AGR.GapObjectOfJsonText`

(6.4-3) does not create such objects.

Note also that the results of `AGR.JsonText`

(6.4-2) do not contain information about dependencies between common subobjects. This is another reason why applying first `AGR.JsonText`

(6.4-2) and then `AGR.GapObjectOfJsonText`

(6.4-3) may yield a **GAP** object with different behaviour.

Applying `AGR.JsonText`

(6.4-2) to a self-referential object such as `[ ~ ]`

will raise a "recursion depth trap" error.

The aim of this JSON interface is to read and write certain data files with **GAP** such that these files become easily accessible independent of **GAP**. The function `AGR.JsonText`

(6.4-2) is intended just as a prototype, variants of this function are very likely to appear in other contexts, for example in order to force certain line formatting or ordering of record components.

It is *not* the aim of the JSON interface to provide self-contained descriptions of arbitrary **GAP** objects, in order to read them into a **GAP** session. Note that those **GAP** objects for which a JSON equivalent exists (and many more) can be easily written to files as they are, and **GAP** can read them efficiently. On the other hand, more complicated **GAP** objects can be written and read via the so-called *pickling*, for which a framework is provided by the **GAP** package **IO** [Neu14].

Here are a few situations which are handled well by pickling but which cannot be addressed with a JSON interface.

Pickling and unpickling take care of common subobjects of the given

**GAP**object. The following example shows that the applying first`AGR.JsonText`

(6.4-2) and then`AGR.GapObjectOfJsonText`

(6.4-3) may yield an object which behaves differently.gap> l:= [ [ 1 ] ];; l[2]:= l[1];; l; [ [ 1 ], [ 1 ] ] gap> new:= AGR.GapObjectOfJsonText( AGR.JsonText( l ) ).value; [ [ 1 ], [ 1 ] ] gap> Add( l[1], 2 ); l; [ [ 1, 2 ], [ 1, 2 ] ] gap> Add( new[1], 2 ); new; [ [ 1, 2 ], [ 1 ] ]

**GAP**admits self-referential objects, for example as follows.gap> l:= [];; l[1]:= l;;

Pickling and unpickling take care of self-referential objects, but

`AGR.JsonText`

(6.4-2) does not support the conversion of such objects.

`‣ AGR.JsonText` ( obj[, mode] ) | ( function ) |

Returns: a new mutable string that describes `obj` as a JSON text, or `fail`

.

If `obj` is a **GAP** object for which a corresponding JSON text exists, according to the mapping described above, then such a JSON text is returned. Otherwise, `fail`

is returned.

If the optional argument `mode` is given and has the value `"ASCII"`

then the result in an ASCII string, otherwise the encoding of strings that are involved in `obj` is kept.

gap> AGR.JsonText( [] ); "[]" gap> AGR.JsonText( "" ); "\"\"" gap> AGR.JsonText( "abc\ndef\cghi" ); "\"abc\\ndef\\u0003ghi\"" gap> AGR.JsonText( rec() ); "{}" gap> AGR.JsonText( [ , 2 ] ); fail gap> str:= [ '\303', '\266' ];; # umlaut o gap> json:= AGR.JsonText( str );; List( json, IntChar ); [ 34, 195, 182, 34 ] gap> AGR.JsonText( str, "ASCII" ); "\"\\u00F6\""

`‣ AGR.GapObjectOfJsonText` ( string ) | ( function ) |

Returns: a new mutable record whose `value`

component, if bound, contains a mutable **GAP** object that represents the JSON text `string`.

If `string` is a string that represents a JSON text then the result is a record with the components `value`

(the corresponding **GAP** object in the sense of the above interface) and `status`

(value `true`

). Otherwise, the result is a record with the components `status`

(value `false`

) and `errpos`

(the position in `string` where the string turns out to be not valid JSON).

gap> AGR.GapObjectOfJsonText( "{ \"a\": 1 }" ); rec( status := true, value := rec( a := 1 ) ) gap> AGR.GapObjectOfJsonText( "{ \"a\": x }" ); rec( errpos := 8, status := false )

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