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# B Examples

### Sections

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 (the case since GAP 4.3); with GAP 4.2 the final table was `semilenlex` 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{}
>                 # 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 (since GAP 4.3, a standard table is a `lenlex` standard table; with GAP 4.2, it was a `semilenlex` 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
March 2016