- Example where ACE is made the Standard Coset Enumerator
- Example of Using ACECosetTableFromGensAndRels
- Example of Using ACE Interactively (Using ACEStart)
- Fun with ACEExample
- 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.

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
*A*_{5}.

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

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 ] ]

Now we illustrate a simple interactive process, with an enumeration of
an index 12 subgroup (isomorphic to *C*_{5}) within *A*_{5}. 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();

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 2^{17} 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
2^{17} 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.

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 *L*_{3}(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