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B Examples

Sections

  1. Example where ACE is made the Standard Coset Enumerator
  2. Example of Using ACECosetTableFromGensAndRels
  3. Example of Using ACE Interactively (Using ACEStart)
  4. Fun with ACEExample
  5. Using ACEReadResearchExample

In this chapter we collect together a number of examples which illustrate the various ways in which the ACE Package may be used, and give some interactions with the ACEExample function. In a number of cases, we have set the InfoLevel of InfoACE to 3, so that all output from ACE is displayed, prepended by ``#I ''. Recall that to also see the commands directed to ACE (behind a ``ToACE> '' prompt), you will need to set the InfoACE level to 4. We have omitted the line

gap> LoadPackage("ace");
true

which is, of course, required at the beginning of any session requiring ACE.

B.1 Example where ACE is made the Standard Coset Enumerator

If ACE is made the standard coset enumerator, one simply uses the method of passing arguments normally used with those commands that invoke CosetTableFromGensAndRels, but one is able to use all options available via the ACE interface. As an example we use ACE to compute the permutation representation of a perfect group from the data library (GAP's perfect group library stores for each group a presentation together with generators of a subgroup as words in the group generators such that the permutation representation on the cosets of this subgroup will be a (nice) faithful permutation representation for the perfect group). The example we have chosen is an extension of a group of order 16 by the simple alternating group A5.

gap> TCENUM:=ACETCENUM;; # Make ACE the standard coset enumerator
gap> G := PerfectGroup(IsPermGroup, 16*60, 1   # Arguments ... as per usual
>                      : max := 50, mess := 10 # ... but we use ACE options
>                      );
A5 2^4
gap> GeneratorsOfGroup(G); # Just to show we indeed have a perm'n rep'n
[ (2,4)(3,5)(7,15)(8,14)(10,13)(12,16), (2,6,7)(3,11,12)(4,14,5)(8,9,13)(10,
    15,16), (1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16), 
  (1,3)(2,8)(4,13)(5,6)(7,10)(9,16)(11,12)(14,15), 
  (1,4)(2,9)(3,13)(5,14)(6,15)(7,11)(8,16)(10,12), 
  (1,5)(2,10)(3,6)(4,14)(7,8)(9,12)(11,16)(13,15) ]
gap> Order(G);
960

The call to PerfectGroup produced an output string that identifies the group G, but we didn't see how ACE became involved here. Let's redo that part of the above example after first setting the InfoLevel of InfoACE to 3, so that we may get to glimpse what's going on behind the scenes.

gap> SetInfoACELevel(3); # Just to see what's going on behind the scenes
gap> # Recall that we did: TCENUM:=ACETCENUM;;
gap> G := PerfectGroup(IsPermGroup, 16*60, 1   # Arguments ... as per usual
>                      : max := 50, mess := 10 # ... but we use ACE options
>                      );
#I  ACE 3.001        Sun Sep 30 22:08:11 2001
#I  =========================================
#I  Host information:
#I    name = rigel
#I  ***
#I    #--- ACE 3.001: Run Parameters ---
#I  Group Name: G;
#I  Group Generators: abstuv;
#I  Group Relators: (a)^2, (s)^2, (t)^2, (u)^2, (v)^2, (b)^3, (st)^2, (uv)^2, 
#I    (su)^2, (sv)^2, (tu)^2, (tv)^2, asau, atav, auas, avat, Bvbu, Bsbvt, 
#I    Bubvu, Btbvuts, (ab)^5;
#I  Subgroup Name: H;
#I  Subgroup Generators: a, b;
#I  Wo:1000000; Max:50; Mess:10; Ti:-1; Ho:-1; Loop:0;
#I  As:0; Path:0; Row:1; Mend:0; No:21; Look:0; Com:10;
#I  C:0; R:0; Fi:11; PMod:3; PSiz:256; DMod:4; DSiz:1000;
#I    #---------------------------------
#I  SG: a=1 r=1 h=1 n=2; l=1 c=+0.00; m=1 t=1
#I  RD: a=11 r=1 h=1 n=12; l=2 c=+0.00; m=11 t=11
#I  RD: a=21 r=2 h=1 n=22; l=2 c=+0.00; m=21 t=21
#I  CC: a=29 r=4 h=1 n=31; l=2 c=+0.00; d=0
#I  CC: a=19 r=4 h=1 n=31; l=2 c=+0.00; d=0
#I  CC: a=19 r=6 h=1 n=36; l=2 c=+0.00; d=0
#I  INDEX = 16 (a=16 r=36 h=1 n=36; l=3 c=0.00; m=30 t=35)
#I  CO: a=16 r=17 h=1 n=17; c=+0.00
#I   coset |      b      B      a      s      t      u      v
#I  -------+-------------------------------------------------
#I       1 |      1      1      1      2      3      4      5
#I       2 |     11     14      4      1      6      8      9
#I       3 |     13     15      5      6      1     10     11
#I       4 |      7      5      2      8     10      1      7
#I       5 |      4      7      3      9     11      7      1
#I       6 |      8     10      7      3      2     12     14
#I       7 |      5      4      6     15     16      5      4
#I       8 |     10      6      8      4     12      2     15
#I       9 |     16     12     10      5     14     15      2
#I      10 |      6      8      9     12      4      3     16
#I      11 |     14      2     11     14      5     16      3
#I      12 |      9     16     15     10      8      6     13
#I      13 |     15      3     13     16     15     14     12
#I      14 |      2     11     16     11      9     13      6
#I      15 |      3     13     12      7     13      9      8
#I      16 |     12      9     14     13      7     11     10
A5 2^4

B.2 Example of Using ACECosetTableFromGensAndRels

The following example calls ACE for up to 800 coset numbers (max := 800) using Mendelsohn style relator processing (mendelsohn) and sets progress messages to be printed every 500 iterations (messages :=500); we do ``SetInfoACELevel(3);'' so that we may see these messages. The value of table, i.e. the GAP coset table, immediately follows the last ACE message (``#I '') line, but both the coset table from ACE and the GAP coset table have been abbreviated. A slightly modified version of this example, which includes the echo option is available on-line via table := ACEExample("perf602p5");. You may wish to peruse the notes in the ACEExample index first, however, by executing ACEExample();. (Note that the final table output here is lenlex standardised.)

gap> SetInfoACELevel(3);           # So we can see the progress messages
gap> G := PerfectGroup(2^5*60, 2);;# See previous example:
gap>                               # "Example where ACE is made the
gap>                               #  Standard Coset Enumerator"
gap> fgens := FreeGeneratorsOfFpGroup(G);;
gap> table := ACECosetTableFromGensAndRels(
>                 # arguments
>                 fgens, RelatorsOfFpGroup(G), fgens{[1]}
>                 # options
>                 : mendelsohn, max:=800, mess:=500);
#I  ACE 3.001        Sun Sep 30 22:10:10 2001
#I  =========================================
#I  Host information:
#I    name = rigel
#I  ***
#I    #--- ACE 3.001: Run Parameters ---
#I  Group Name: G;
#I  Group Generators: abstuvd;
#I  Group Relators: (s)^2, (t)^2, (u)^2, (v)^2, (d)^2, aad, (b)^3, (st)^2, 
#I    (uv)^2, (su)^2, (sv)^2, (tu)^2, (tv)^2, Asau, Atav, Auas, Avat, Bvbu, 
#I    dAda, dBdb, (ds)^2, (dt)^2, (du)^2, (dv)^2, Bubvu, Bsbdvt, Btbvuts, 
#I    (ab)^5;
#I  Subgroup Name: H;
#I  Subgroup Generators: a;
#I  Wo:1000000; Max:800; Mess:500; Ti:-1; Ho:-1; Loop:0;
#I  As:0; Path:0; Row:1; Mend:1; No:28; Look:0; Com:10;
#I  C:0; R:0; Fi:13; PMod:3; PSiz:256; DMod:4; DSiz:1000;
#I    #---------------------------------
#I  SG: a=1 r=1 h=1 n=2; l=1 c=+0.00; m=1 t=1
#I  RD: a=321 r=68 h=1 n=412; l=5 c=+0.00; m=327 t=411
#I  CC: a=435 r=162 h=1 n=719; l=9 c=+0.00; d=0
#I  CL: a=428 r=227 h=1 n=801; l=13 c=+0.00; m=473 t=800
#I  DD: a=428 r=227 h=1 n=801; l=14 c=+0.00; d=33
#I  CO: a=428 r=192 h=243 n=429; l=15 c=+0.00; m=473 t=800
#I  INDEX = 480 (a=480 r=210 h=484 n=484; l=18 c=0.00; m=480 t=855)
#I  CO: a=480 r=210 h=481 n=481; c=+0.00
#I   coset |      a      A      b      B      s      t      u      v      d
#I  -------+---------------------------------------------------------------
#I       1 |      1      1      7      6      2      3      4      5      1
#I       2 |      4      4     22     36      1      8     10     11      2
... 476 lines omitted here ...
#I     479 |    479    479    384    383    475    468    470    471    479
#I     480 |    480    480    421    420    470    469    475    476    480
[ [ 1, 8, 13, 6, 7, 4, 5, 2, 34, 35, 32, 33, 3, 48, 49, 46, 47, 57, 59, 28, 
      21, 25, 62, 64, 22, 26, 66, 20, 67, 69, 74, 11, 12, 9, 10, 89, 65, 87, 
... 30 lines omitted here ...
      477, 438, 478, 446, 475, 479, 471, 473, 476, 469 ], 
  [ 1, 8, 13, 6, 7, 4, 5, 2, 34, 35, 32, 33, 3, 48, 49, 46, 47, 57, 59, 28, 
      21, 25, 62, 64, 22, 26, 66, 20, 67, 69, 74, 11, 12, 9, 10, 89, 65, 87, 
... 30 lines omitted here ...
      477, 438, 478, 446, 475, 479, 471, 473, 476, 469 ], 
... 363 lines omitted here ...
  [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 
      21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 
... 30 lines omitted here ...
      472, 473, 474, 475, 476, 477, 478, 479, 480 ] ]

B.3 Example of Using ACE Interactively (Using ACEStart)

Now we illustrate a simple interactive process, with an enumeration of an index 12 subgroup (isomorphic to C5) within A5. Observe that we have relied on the default level of messaging from ACE (messages = 0) which gives a result line (the ``#I INDEX'' line here) only, without parameter information. The result line is visible in the Info-ed component of the output below because we set the InfoLevel of InfoACE to a value of at least 2 (in fact we set it to 3; doing ``SetInfoACELevel(2);'' would make only the result line visible). We have however used the option echo, so that we can see how the interface handled the arguments and options. On-line try: SetInfoACELevel(3); ACEExample("A5-C5", ACEStart); (this is nearly equivalent to the sequence following, but the variables F, a, b, G are not accessible, being ``local'' to ACEExample).

gap> SetInfoACELevel(3); # So we can see output from ACE binary
gap> F := FreeGroup("a","b");; a := F.1;;  b := F.2;;
gap> G := F / [a^2, b^3, (a*b)^5 ];
<fp group on the generators [ a, b ]>
gap> ACEStart(FreeGeneratorsOfFpGroup(G), RelatorsOfFpGroup(G), [a*b]
>          # Options
>          : echo, # Echo handled by GAP (not ACE)
>            enum := "A_5",  # Give the group G a meaningful name
>            subg := "C_5"); # Give the subgroup a meaningful name
ACEStart called with the following arguments:
 Group generators : [ a, b ]
 Group relators : [ a^2, b^3, a*b*a*b*a*b*a*b*a*b ]
 Subgroup generators : [ a*b ]
#I  ACE 3.001        Sun Sep 30 22:11:42 2001
#I  =========================================
#I  Host information:
#I    name = rigel
ACEStart called with the following options:
 echo := true (not passed to ACE)
 enum := A_5
 subg := C_5
#I  ***
#I  INDEX = 12 (a=12 r=16 h=1 n=16; l=3 c=0.00; m=14 t=15)
1

The return value on the last line is an ``index'' that identifies the interactive process; we use this ``index'' with functions that need to interact with the interactive ACE process; we now demonstrate this with the interactive version of ACEStats:

gap> ACEStats(1);
rec( index := 12, cputime := 0, cputimeUnits := "10^-2 seconds", 
  activecosets := 12, maxcosets := 14, totcosets := 15 )
gap> # Actually, we didn't need to pass an argument to ACEStats()
gap> # ... we could have relied on the default:
gap> ACEStats();
rec( index := 12, cputime := 0, cputimeUnits := "10^-2 seconds", 
  activecosets := 12, maxcosets := 14, totcosets := 15 )

Similarly, we may use ACECosetTable with 0 or 1 arguments, which is the interactive version of ACECosetTableFromGensAndRels, and which returns a standard table.

gap> ACECosetTable(); # Interactive version of ACECosetTableFromGensAndRels()
#I  CO: a=12 r=13 h=1 n=13; c=+0.00
#I   coset |      b      B      a
#I  -------+---------------------
#I       1 |      3      2      2
#I       2 |      1      3      1
#I       3 |      2      1      4
#I       4 |      8      5      3
#I       5 |      4      8      6
#I       6 |      9      7      5
#I       7 |      6      9      8
#I       8 |      5      4      7
#I       9 |      7      6     10
#I      10 |     12     11      9
#I      11 |     10     12     12
#I      12 |     11     10     11
[ [ 2, 1, 4, 3, 7, 8, 5, 6, 10, 9, 12, 11 ], 
  [ 2, 1, 4, 3, 7, 8, 5, 6, 10, 9, 12, 11 ], 
  [ 3, 1, 2, 5, 6, 4, 8, 9, 7, 11, 12, 10 ], 
  [ 2, 3, 1, 6, 4, 5, 9, 7, 8, 12, 10, 11 ] ]
gap> # To terminate the interactive process we do:
gap> ACEQuit(1); # Again, we could have omitted the 1
gap> # If we had more than one interactive process we could have
gap> # terminated them all in one go with:
gap> ACEQuitAll();

B.4 Fun with ACEExample

First let's see the ACEExample index (obtained with no argument, with "index" as argument, or with a non-existent example as argument):

gap> ACEExample();
#I                   ACEExample Index (Table of Contents)
#I                   ------------------------------------
#I  This table of possible examples is displayed when calling ACEExample 
#I  with no arguments, or with the argument: "index" (meant in the sense
#I  of `list'), or with a non-existent example name.
#I  
#I  The following ACE examples are available (in each case, for a subgroup
#I  H of a group G, the cosets of H in G are enumerated):
#I  
#I    Example          G                      H              strategy
#I    -------          -                      -              --------
#I    "A5"             A_5                    Id             default
#I    "A5-C5"          A_5                    C_5            default
#I    "C5-fel0"        C_5                    Id             felsch := 0
#I    "F27-purec"      F(2,7) = C_29          Id             purec
#I    "F27-fel0"       F(2,7) = C_29          Id             felsch := 0
#I    "F27-fel1"       F(2,7) = C_29          Id             felsch := 1
#I    "M12-hlt"        M_12 (Matthieu group)  Id             hlt
#I    "M12-fel1"       M_12 (Matthieu group)  Id             felsch := 1
#I    "SL219-hard"     SL(2,19)               |G : H| = 180  hard
#I    "perf602p5"      PerfectGroup(60*2^5,2) |G : H| = 480  default
#I  * "2p17-fel1"      |G| = 2^17             Id             felsch := 1
#I    "2p17-fel1a"     |G| = 2^17             |G : H| = 1    felsch := 1
#I    "2p17-2p3-fel1"  |G| = 2^17             |G : H| = 2^3  felsch := 1
#I    "2p17-2p14-fel1" |G| = 2^17             |G : H| = 2^14 felsch := 1
#I    "2p17-id-fel1"   |G| = 2^17             Id             felsch := 1
#I  * "2p18-fel1"      |G| = 2^18             |G : H| = 2    felsch := 1
#I  * "big-fel1"       |G| = 2^18.3           |G : H| = 6    felsch := 1
#I  * "big-hard"       |G| = 2^18.3           |G : H| = 6    hard
#I  
#I  Notes
#I  -----
#I  1. The example (first) argument of ACEExample() is a string; each
#I     example above is in double quotes to remind you to include them.
#I  2. By default, ACEExample applies ACEStats to the chosen example. You 
#I     may alter the ACE function used, by calling ACEExample with a 2nd 
#I     argument; choose from: ACECosetTableFromGensAndRels (or, equival-
#I     ently ACECosetTable), or ACEStart, e.g. `ACEExample("A5", ACEStart);'
#I  3. You may call ACEExample with additional ACE options (entered after a
#I     colon in the usual way for options), e.g. `ACEExample("A5" : hlt);' 
#I  4. Try the *-ed examples to explore how to modify options when an
#I     enumeration fails (just follow the instructions you get within the
#I     break-loop, or see Notes 2. and 3.).
#I  5. Try `SetInfoACELevel(3);' before calling ACEExample, to see the
#I     effect of setting the "mess" (= "messages") option.
#I  6. To suppress a long output, use a double semicolon (`;;') after the
#I     ACEExample command. (However, this does not suppress Info-ed output.)
#I  7. Also, try `SetInfoACELevel(2);' or `SetInfoACELevel(4);' before 
#I     calling ACEExample.
gap> 

Notice that the example we first met in Section Using ACE Directly to Generate a Coset Table, the Fibonacci group F(2,7), is available via examples "F27-purec", "F27-fel0", and "F27-fel1" (with 2nd argument ACECosetTableFromGensAndRels to produce a coset table), except that each of these enumerate the cosets of its trivial subgroup (of index 29). Let's experiment with the first of these F(2,7) examples; since this example uses the messages option, we ought to set the InfoLevel of InfoACE to 3, first, but since the coset table is quite long, we will be content for the moment with applying the default function ACEStats to the example.

Before exhibiting the example we list a few observations that should be made. Observe that the first group of Info lines list the commands that are executed; these lines are followed by the result of the echo option (see option echo); which in turn are followed by Info messages from ACE courtesy of the non-zero value of the messages option (and we see these because we first set the InfoLevel of InfoACE to 3); and finally, we get the output (record) of the ACEStats command.

Observe also that ACE uses the same generators as are input; this will always occur if you stick to single lowercase letters for your generator names. Note, also that capitalisation is used by ACE as a short-hand for inverses, e.g. C = c^-1 (see Group Relators in the ACE ``Run Parameters'' block).

gap> SetInfoACELevel(3);
gap> ACEExample("F27-purec");
#I  # ACEExample "F27-purec" : enumeration of cosets of H in G,
#I  # where G = F(2,7) = C_29, H = Id, using purec strategy.
#I  #
#I  # F, G, a, b, c, d, e, x, y are local to ACEExample
#I  # We define F(2,7) on 7 generators
#I  F := FreeGroup("a","b","c","d","e", "x", "y"); 
#I       a := F.1;  b := F.2;  c := F.3;  d := F.4; 
#I       e := F.5;  x := F.6;  y := F.7;
#I  G := F / [a*b*c^-1, b*c*d^-1, c*d*e^-1, d*e*x^-1, 
#I            e*x*y^-1, x*y*a^-1, y*a*b^-1];
#I  ACEStats(
#I      FreeGeneratorsOfFpGroup(G), 
#I      RelatorsOfFpGroup(G), 
#I      [] # Generators of identity subgroup (empty list)
#I      # Options that don't affect the enumeration
#I      : echo, enum := "F(2,7), aka C_29", subg := "Id", 
#I      # Other options
#I      wo := "2M", mess := 25000, purec);
ACEStats called with the following arguments:
 Group generators : [ a, b, c, d, e, x, y ]
 Group relators : [ a*b*c^-1, b*c*d^-1, c*d*e^-1, d*e*x^-1, e*x*y^-1, 
  x*y*a^-1, y*a*b^-1 ]
 Subgroup generators : [  ]
#I  ACE 3.001        Sun Sep 30 22:16:08 2001
#I  =========================================
#I  Host information:
#I    name = rigel
ACEStats called with the following options:
 echo := true (not passed to ACE)
 enum := F(2,7), aka C_29
 subg := Id
 wo := 2M
 mess := 25000
 purec (no value, passed to ACE via option: pure c)
#I  ***
#I    #--- ACE 3.001: Run Parameters ---
#I  Group Name: F(2,7), aka C_29;
#I  Group Generators: abcdexy;
#I  Group Relators: abC, bcD, cdE, deX, exY, xyA, yaB;
#I  Subgroup Name: Id;
#I  Subgroup Generators: ;
#I  Wo:2M; Max:142855; Mess:25000; Ti:-1; Ho:-1; Loop:0;
#I  As:0; Path:0; Row:0; Mend:0; No:0; Look:0; Com:100;
#I  C:1000; R:0; Fi:1; PMod:0; PSiz:256; DMod:4; DSiz:1000;
#I    #---------------------------------
#I  DD: a=5290 r=1 h=1050 n=5291; l=8 c=+0.00; d=2
#I  CD: a=10410 r=1 h=2149 n=10411; l=13 c=+0.01; m=10410 t=10410
#I  DD: a=15428 r=1 h=3267 n=15429; l=18 c=+0.01; d=0
#I  DD: a=20430 r=1 h=4386 n=20431; l=23 c=+0.02; d=1
#I  DD: a=25397 r=1 h=5519 n=25399; l=28 c=+0.01; d=1
#I  CD: a=30313 r=1 h=6648 n=30316; l=33 c=+0.01; m=30313 t=30315
#I  DS: a=32517 r=1 h=7326 n=33240; l=36 c=+0.01; s=2000 d=997 c=4
#I  DS: a=31872 r=1 h=7326 n=33240; l=36 c=+0.00; s=4000 d=1948 c=53
#I  DS: a=29077 r=1 h=7326 n=33240; l=36 c=+0.00; s=8000 d=3460 c=541
#I  DS: a=23433 r=1 h=7326 n=33240; l=36 c=+0.01; s=16000 d=5940 c=2061
#I  DS: a=4163 r=1 h=7326 n=33240; l=36 c=+0.03; s=32000 d=447 c=15554
#I  INDEX = 29 (a=29 r=1 h=33240 n=33240; l=37 c=0.15; m=33237 t=33239)
rec( index := 29, cputime := 15, cputimeUnits := "10^-2 seconds", 
  activecosets := 29, maxcosets := 33237, totcosets := 33239 )

Now let's see that we can add some new options, even ones that over-ride the example's options; but first we'll reduce the output a bit by setting the InfoLevel of InfoACE to 2 and since we are not going to observe any progress messages from ACE with that InfoACE level we'll set messages := 0; also we'll use the function ACECosetTableFromGensAndRels and so it's like our first encounter with F(2,7) we'll add the subgroup generator c via sg := ["c"] (see option sg). Observe that "c" is a string not a GAP group generator; to convert a list of GAP words sgens in generators fgens, suitable for an assignment of the sg option use the construction: ToACEWords(fgens, sgens) (see ToACEWords). Note again that if only single lowercase letter strings are used to identify the GAP group generators, the same strings are used to identify those generators in ACE. (It's actually fortunate that we could pass the value of sg as a string here, since the generators of each ACEExample example are local variables and so are not accessible, though we could call ACEExample with 2nd argument ACEStart and use ACEGroupGenerators to get at them.) For good measure, we also change the string identifying the subgroup (since it will no longer be the trivial group), via the subgroup option (see option subgroup).

In considering the example following, observe that in the Info block all the original example options are listed along with our new options sg := [ "c" ], messages := 0 after the tag ``# User Options''. Following the Info block there is a block due to echo; in its listing of the options first up there is aceexampleoptions alerting us that we passed some ACEExample options; observe also that in this block subg := Id and mess := 25000 disappear (they are over-ridden by subgroup := "< c >", messages := 0, but the quotes for the value of subgroup are not visible); note that we don't have to use the same abbreviations for options to over-ride them. Also observe that our new options are last.

gap> SetInfoACELevel(2);
gap> ACEExample("F27-purec", ACECosetTableFromGensAndRels
>               : sg := ["c"], subgroup := "< c >", messages := 0);
#I  # ACEExample "F27-purec" : enumeration of cosets of H in G,
#I  # where G = F(2,7) = C_29, H = Id, using purec strategy.
#I  #
#I  # F, G, a, b, c, d, e, x, y are local to ACEExample
#I  # We define F(2,7) on 7 generators
#I  F := FreeGroup("a","b","c","d","e", "x", "y"); 
#I       a := F.1;  b := F.2;  c := F.3;  d := F.4; 
#I       e := F.5;  x := F.6;  y := F.7;
#I  G := F / [a*b*c^-1, b*c*d^-1, c*d*e^-1, d*e*x^-1, 
#I            e*x*y^-1, x*y*a^-1, y*a*b^-1];
#I  ACECosetTableFromGensAndRels(
#I      FreeGeneratorsOfFpGroup(G), 
#I      RelatorsOfFpGroup(G), 
#I      [] # Generators of identity subgroup (empty list)
#I      # Options that don't affect the enumeration
#I      : echo, enum := "F(2,7), aka C_29", subg := "Id", 
#I      # Other options
#I      wo := "2M", mess := 25000, purec, 
#I      # User Options
#I        sg := [ "c" ],
#I        subgroup := "< c >",
#I        messages := 0);
ACECosetTableFromGensAndRels called with the following arguments:
 Group generators : [ a, b, c, d, e, x, y ]
 Group relators : [ a*b*c^-1, b*c*d^-1, c*d*e^-1, d*e*x^-1, e*x*y^-1, 
  x*y*a^-1, y*a*b^-1 ]
 Subgroup generators : [  ]
ACECosetTableFromGensAndRels called with the following options:
 aceexampleoptions := true (inserted by ACEExample, not passed to ACE)
 echo := true (not passed to ACE)
 enum := F(2,7), aka C_29
 wo := 2M
 purec (no value, passed to ACE via option: pure c)
 sg := [ "c" ] (brackets are not passed to ACE)
 subgroup := < c >
 messages := 0
#I  INDEX = 1 (a=1 r=2 h=2 n=2; l=4 c=0.00; m=332 t=332)
[ [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], 
  [ 1 ], [ 1 ], [ 1 ], [ 1 ] ]

Now following on from our last example we shall demonstrate how one can recover from a break-loop (see Section Using ACE Directly to Generate a Coset Table). To force the break-loop we pass max := 2 (see option max), while using the ACE interface function ACECosetTableFromGensAndRels with ACEExample; in this way, ACE will not be able to complete the enumeration, and hence enters a break-loop when it tries to provide a complete coset table. While we're at it we'll pass the hlt (see option hlt) strategy option (which will over-ride purec). (The InfoACE level is still 2.) To avoid getting a trace-back during the break-loop (which can look a little scary to the unitiated) we will set OnBreak (see OnBreak) as follows:

gap> NormalOnBreak := OnBreak;; # Save the old value to restore it later
gap> OnBreak := function() Where(0); end;;

Note that there are some ``user-input'' comments inserted at the brk> prompt.

gap> ACEExample("F27-purec", ACECosetTableFromGensAndRels
>               : sg := ["c"], subgroup := "< c >", max := 2, hlt);
#I  # ACEExample "F27-purec" : enumeration of cosets of H in G,
#I  # where G = F(2,7) = C_29, H = Id, using purec strategy.
#I  #
#I  # F, G, a, b, c, d, e, x, y are local to ACEExample
#I  # We define F(2,7) on 7 generators
#I  F := FreeGroup("a","b","c","d","e", "x", "y"); 
#I       a := F.1;  b := F.2;  c := F.3;  d := F.4; 
#I       e := F.5;  x := F.6;  y := F.7;
#I  G := F / [a*b*c^-1, b*c*d^-1, c*d*e^-1, d*e*x^-1, 
#I            e*x*y^-1, x*y*a^-1, y*a*b^-1];
#I  ACECosetTableFromGensAndRels(
#I      FreeGeneratorsOfFpGroup(G), 
#I      RelatorsOfFpGroup(G), 
#I      [] # Generators of identity subgroup (empty list)
#I      # Options that don't affect the enumeration
#I      : echo, enum := "F(2,7), aka C_29", subg := "Id", 
#I      # Other options
#I      wo := "2M", mess := 25000, purec, 
#I      # User Options
#I        sg := [ "c" ],
#I        subgroup := "< c >",
#I        max := 2,
#I        hlt := true);
ACECosetTableFromGensAndRels called with the following arguments:
 Group generators : [ a, b, c, d, e, x, y ]
 Group relators : [ a*b*c^-1, b*c*d^-1, c*d*e^-1, d*e*x^-1, e*x*y^-1, 
  x*y*a^-1, y*a*b^-1 ]
 Subgroup generators : [  ]
ACECosetTableFromGensAndRels called with the following options:
 aceexampleoptions := true (inserted by ACEExample, not passed to ACE)
 echo := true (not passed to ACE)
 enum := F(2,7), aka C_29
 wo := 2M
 mess := 25000
 purec (no value, passed to ACE via option: pure c)
 sg := [ "c" ] (brackets are not passed to ACE)
 subgroup := < c >
 max := 2
 hlt (no value)
#I  OVERFLOW (a=2 r=1 h=1 n=3; l=4 c=0.00; m=2 t=2)
Error, no coset table ...
 the `ACE' coset enumeration failed with the result:
 OVERFLOW (a=2 r=1 h=1 n=3; l=4 c=0.00; m=2 t=2)
Entering break read-eval-print loop ...
 try relaxing any restrictive options
 e.g. try the `hard' strategy or increasing `workspace'
 type: '?strategy options' for info on strategies
 type: '?options for ACE' for info on options
 type: 'DisplayACEOptions();' to see current ACE options;
 type: 'SetACEOptions(:<option1> := <value1>, ...);'
 to set <option1> to <value1> etc.
 (i.e. pass options after the ':' in the usual way)
 ... and then, type: 'return;' to continue.
 Otherwise, type: 'quit;' to quit to outer loop.
brk> # Let's give ACE enough coset numbers to work with ...
brk> # and while we're at it see the effect of 'echo := 2' :
brk> SetACEOptions(: max := 0, echo := 2);
brk> # Let's check what the options are now:
brk> DisplayACEOptions();
rec(
  enum := "F(2,7), aka C_29",
  wo := "2M",
  mess := 25000,
  purec := true,
  sg := [ "c" ],
  subgroup := "< c >",
  hlt := true,
  max := 0,
  echo := 2 )

brk> # That's ok ... so now we 'return;' to escape the break-loop
brk> return;
ACECosetTableFromGensAndRels called with the following arguments:
 Group generators : [ a, b, c, d, e, x, y ]
 Group relators : [ a*b*c^-1, b*c*d^-1, c*d*e^-1, d*e*x^-1, e*x*y^-1, 
  x*y*a^-1, y*a*b^-1 ]
 Subgroup generators : [  ]
ACECosetTableFromGensAndRels called with the following options:
 enum := F(2,7), aka C_29
 wo := 2M
 mess := 25000
 purec (no value, passed to ACE via option: pure c)
 sg := [ "c" ] (brackets are not passed to ACE)
 subgroup := < c >
 hlt (no value)
 max := 0
 echo := 2 (not passed to ACE)
Other options set via ACE defaults:
 asis := 0
 compaction := 10
 ct := 0
 dmode := 0
 dsize := 1000
 fill := 1
 hole := -1
 lookahead := 1
 loop := 0
 mendelsohn := 0
 no := 0
 path := 0
 pmode := 0
 psize := 256
 row := 1
 rt := 1000
 time := -1
#I  INDEX = 1 (a=1 r=2 h=2 n=2; l=3 c=0.00; m=2049 t=3127)
[ [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], 
  [ 1 ], [ 1 ], [ 1 ], [ 1 ] ]

Observe that purec did not disappear from the option list; nevertheless, it is over-ridden by the hlt option (at the ACE level). Observe the ``Other options set via ACE defaults'' list of options that has resulted from having the echo := 2 option (see option echo). Observe, also, that hlt is nowhere near as good, here, as purec (refer to Section Using ACE Directly to Generate a Coset Table): whereas purec completed the same enumeration with a total number of coset numbers of 332, the hlt strategy required 3127.

Before we finish this section, let us say something about the examples listed when one calls ACEExample with no arguments that have a * beside them; these are examples for which the enumeration fails to complete. The first such example listed is "2p17-fel1", where a group of order 217 is enumerated over the identity subgroup with the felsch := 1 strategy. The enumeration fails after defining a total number of 416664 coset numbers. (In fact, the enumeration can be made to succeed by simply increasing workspace to "4700k", but in doing so a total of 783255 coset numbers are defined.) With the example "2p17-fel1a" the same group is again enumerated, again with the felsch := 1 strategy, but this time over the group itself and after tweaking a few options, to see how well we can do. The other "2p17-XXX" examples are again enumerations of the same group but over smaller and smaller subgroups, until we once again enumerate over the identity subgroup but far more efficiently this time (only needing to define a total of 550659 coset numbers, which can be achieved with workspace set to "3300k").

The other *-ed examples enumerate overgroups of the group of order 217 of the "2p17-XXX" examples. It's recommended that you try these with second argument ACECosetTableFromGensAndRels so that you enter a break-loop, where you can experiment with modifying the options using SetACEOptions. The example "2p18-fel1" can be made to succeed with hard, mend, workspace := "10M"; why don't you see if you can do better! There are no hints for the other two *-ed examples; they are exercises for you to try.

Let's now restore the original value of OnBreak:

gap> OnBreak := NormalOnBreak;;

Of course, running ACEExample with ACEStart as second argument opens up far more flexibility. Try it! Have fun! Play with as many options as you can.

B.5 Using ACEReadResearchExample

Without an argument, the function ACEReadResearchExample reads the file "pgrelfind.g" in the res-examples directory which defines GAP functions such as PGRelFind, that were used in CHHR01 to show that the group L3(5) has deficiency 0.

The deficiency of a finite presentation {X | R} of a finite group G is |R| − |X|, and the deficiency of the group G is the minimum of the deficiencies of all finite presentations of G.

Let us now invoke ACEReadResearchExample with no arguments:

gap> ACEReadResearchExample();
#I  The following are now defined:
#I  
#I  Functions:
#I    PGRelFind, ClassesGenPairs, TranslatePresentation,
#I    IsACEResExampleOK
#I  Variables:
#I    ACEResExample, ALLRELS, newrels, F, a, b, newF, x, y,
#I    L2_8, L2_16, L3_3s, U3_3s, M11, M12, L2_32,
#I    U3_4s, J1s, L3_5s, PSp4_4s, presentations
#I  
#I  Also:
#I  
#I  TCENUM = ACETCENUM  (Todd-Coxeter Enumerator is now ACE)
#I  
#I  For an example of their application, you might like to try:
#I  gap> ACEReadResearchExample("doL28.g" : optex := [1,2,4,5,8]);
#I  (the output is 65 lines followed by a 'gap> ' prompt)
#I  
#I  For information type: ?Using ACEReadResearchExample
gap> 

The output (Info-ed at InfoACELevel 1) states that a number of new functions are defined. During a GAP session, you can find out details of these functions by typing:

gap> ?Using ACEReadResearchExample

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ACE manual
February 2020