%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %W examples.tex ACE appendix - examples Alexander Hulpke %W Joachim Neub"user %W Greg Gamble %% %Y Copyright (C) 2000 Centre for Discrete Mathematics and Computing %Y Department of Information Tech. & Electrical Eng. %Y University of Queensland, Australia. %%
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
\beginexample
gap> LoadPackage("ace");
true
\endexample
which is, of course, required at the beginning of any session
requiring {\ACE}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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
$A_5$.
\beginexample
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
\endexample
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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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 ```SetInfoLevel(InfoACE, 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.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Example of Using ACE Interactively (Using ACEStart)}
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 ```SetInfoLevel(InfoACE, 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:
`SetInfoLevel(InfoACE, 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').
\beginexample
gap> SetInfoLevel(InfoACE, 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
\endexample
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':
\beginexample
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 )
\endexample
Similarly, we may use `ACECosetTable' with 0 or 1 arguments, which is
the interactive version of `ACECosetTableFromGensAndRels', and which
returns a standard table.
\beginexample
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();
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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):
\beginexample
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 `SetInfoLevel(InfoACE, 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 `SetInfoLevel(InfoACE, 2);' or `SetInfoLevel(InfoACE, 4);'
#I before calling ACEExample.
gap>
\endexample
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).
\beginexample
gap> SetInfoLevel(InfoACE, 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 )
\endexample
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*.
\beginexample
gap> SetInfoLevel(InfoACE, 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 ] ]
\endexample
\atindex{break-loop}{@\noexpand`break'-loop}\indextt{OnBreak}
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~"ref:OnBreak") as follows:
\beginexample
gap> NormalOnBreak := OnBreak;; # Save the old value to restore it later
gap> OnBreak := function() Where(0); end;;
\endexample
Note that there are some ``user-input'' comments inserted at the
`brk>' prompt.
\beginexample
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(: := , ...);'
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 ] ]
\endexample
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-"' 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-"' 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':
\beginexample
gap> OnBreak := NormalOnBreak;;
\endexample
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.
\indextt{PGRelFind}
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 \cite{CHHR01}
to show that the group $L_3(5)$ has deficiency 0. %display{nonhtml} % %*Some background* %enddisplay
The *deficiency* of a finite presentation $\{X \mid R\}$ 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 $G$.
%display{nonhtml} %The following result (with a constructive proof) is Lemma 2.2 in %\cite{CHHR01}. % %*Lemma* %Let $G$ be a simple group with trivial Schur multiplier. Suppose $G$ %has a presentation of the form %$$ %P = \{ a, b \mid a^2=1, b^p=1, w(a,b)=1\} %$$ %with $p$ prime. Then $G$ has deficiency zero. % %Prior to the paper \cite{CHHR01} there remained just two simple groups %with trivial Schur multiplier of order less than one million, namely %$L_3(5)$ and $PSp_4(4)$, that had not been shown to have deficiency %zero. Using `TranslatePresentation' and `PGRelFind' the question for %$L_3(5)$ is now settled, but whether $PSp_4(4)$ has deficiency 0 is %still open. % %We now sketch the method used in \cite{CHHR01} to show that a simple %group $G$ with trivial Schur multiplier (in particular, $L_3(5)$) has %deficiency 0. We start with a known two-generator presentation of the %group $G$, on say $x$ and $y$. Then an invocation of %`TranslatePresentation' (which uses Tietze transformations) translates %the presentation into a presentation on two new generators $a$ and $b$ %say, having several relators but for which the first two relators are %of form $a^2$ and $b^p$ for some prime $p$. Finally, `PGRelFind' is %invoked to find a presentation on the generators $a$ and $b$ found by %`TranslatePresentation' that contains $a^2$ and $b^p$ and just one %more relator. If the search by `PGRelFind' is successful then the %Lemma shows the group has deficiency zero. % %*Some details (what the file `pgrelfind.g' defines)* %enddisplay
Let us now invoke `ACEReadResearchExample' with no arguments:
\beginexample
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>
\endexample
%display{nontext}
The output (`Info'-ed at `InfoACE' level 1) states that a number of new
functions are defined. During a {\GAP} session, you can find out
details of these functions by typing:
%enddisplay %display{nonhtml} %The output (`Info'-ed at `InfoACE' level 1) states that a number of %functions and variables have been defined, which we will now describe. % %\>PGRelFind( <fgens>, <rels>, <sgens> ) F % %For the perfect *simple* group $S = \langle <fgens> \mid <rels> %\rangle$ with subgroup $\langle <sgens> \rangle$ where each element of %the list <sgens> is a word in the generators <fgens>, `PGRelFind' %tries to find a third relator such that together with the two %order-defining relators of <fgens> a presentation on three relators is %determined, and if successful returns a record with fields `fgens := %<fgens>', `rels' set to the new 3-relator list found, and `sgens := %<sgens>'. <fgens> should be a list of two generators of a free group, %say `[<a>, <b>]'; <rels> should contain words in the generators %<fgens>, the first of which should determine that <a> is an %involution, and the second should determine that <b> is of order <p> %for some prime <p>. The subgroup $\langle <sgens> \rangle$ should %ideally be a maximal subgroup of $\langle <fgens> \mid <rels> %\rangle$. It turns out that one only needs to test for relators that %are the products of an odd number of bi-syllables of form $<a><b>^k$, %for some integer $k$ in the range $1,\ldots,<p> - 1$. We will use the %term *bi-syllabic length* to mean the number of bi-syllables of form %$<a><b>^k$ in such a relator. Furthermore, each relator we test can be %assumed to have the prefix (which we subsequently call the <head>) %$<a><b><a><b><a><b>^{-1}$ if <p> is 3, or $<a><b>$ otherwise. % %Each relator tested is of form `<head> * <middle> * <tail>', where %each of <head>, <middle>, <tail> is a word of integral bi-syllabic %length. Moreover, % %\beginlist%unordered % %\item{--} <head> is fixed (as mentioned above); % %\item{--} all <tail>s of the opposite parity bi-syllabic length up to %a maximum prescribed length are pre-computed (since <head> is %generally of odd bi-syllabic length, <tail>s are generally of even %bi-syllabic length); and % %\item{--} the bi-syllabic length of the <middle>s is always even, %starting from an initial minimum <middle> length and increasing in %steps of `<granularity> + 2', where <granularity> is defined to be the %difference of the maximum and minimum bi-syllabic lengths of a <tail> %(see below). % %\endlist % %Since the <tail>s in general are not all of the same length, it is %possible when a search is successful that the first relator found is %not the shortest. Hence searching continues until a relator of %shortest length can be guaranteed. % %To provide some user control of the algorithm, `PGRelFind' has a %number of options (entered after the arguments and after a colon in %the usual way): % %\beginitems % %\quad`head := <head>' & %Redefines <head> which by default is `<a>*<b>*<a>*<b>*<a>/<b>' if the %order of <b> is 3, or `<a>*<b>' otherwise, where <a> and <b> are as %defined above; <head> must be a word consisting of `<a>*<b>^<k>' %bi-syllables for integers <k>. % %\quad`Nrandom := <nOrFunc>' & %Sets the number of <middle>s to be generated for each length of a %<middle>; <nOrFunc> may be a non-negative integer or a single-argument %function that returns a positive integer. % %& %Each <middle> may be generated sequentially or randomly. If `<nOrFunc> %= 0' then <middle>s are generated sequentially (and hence %exhaustively), as they are by default. If <nOrFunc> is a positive %integer then, for each bi-syllabic length of a <middle>, <nOrFunc> %``random'' <middle>s are generated. If <nOrFunc> is a single-argument %function then for each bi-syllabic length <len> of a <middle>, %`<nOrFunc>(<len>)' ``random'' <middle>s are generated. % %\quad`ACEworkspace := <workspace>' & %Sets the option `workspace' used by {\ACE} to <workspace> when running %the index check of the large subgroup. If the index is right, {\ACE} %is set to use `2*<workspace>' to check the index of the cyclic %subgroup $\langle <b> \rangle$. By default, <workspace> is set to %`10^6', which is too small for many of the `do<XXX>.g' files (see %below). Of course, groups for which {\ACE} overflows the workspace are %indistinguishable from infinite groups, and one can't easily be sure %whether a relator wasn't found because the workspace was too small or %because there wasn't one to be found. % %\quad`Ntails := <Ntails>' & %Sets the approximate number of <tail>s generated to <Ntails>; it is %used to set `maxTailLength' (described next); <Ntails> must a positive %integer. The default behaviour is given by `<Ntails> = 2048'. % %\quad`maxTailLength := <len>' & %Sets the *intended* maximum bi-syllabic length of a <tail> to <len>; %<len> must be a positive integer. % %& %The default behaviour is given by `<len> = LogInt(<Ntails>, <orderb> - %1)', where <orderb> is the order of <b>. The *actual* maximum %bi-syllabic length of a <tail> may be set 1 less, in order that it has %the same parity as the minimum bi-syllabic length of a <tail>. % %\quad`minMiddleLength := <len>' & %Sets the minimum bi-syllabic length of a <middle> (which, by default %is 0) to <len>; <len> should be a non-negative even integer. % %\quad`maxMiddleLength := <len>' & %Sets the *intended* maximum bi-syllabic length of a <middle> to <len>; %<len> should be a positive integer. % %& %The default behaviour is given by `<len> = 30'. The *actual* maximum %bi-syllabic length of a <middle> is adjusted downwards by the least %amount necessary to ensure that the difference of the minimum and %maximum bi-syllabic lengths of a <middle> is divisible by %`<granularity> + 2'. (Recall that <granularity> is the difference of %the maximum and minimum bi-syllabic lengths of a <tail>.) % %\enditems % %Please note that the relators tested are saved in the lists `ALLRELS' %and `newrels' (see~"ALLRELS"). % %\>ClassesGenPairs( <G>, <orderx>, <ordery> ) F % %finds, and returns as a list of 2-element lists, generator pairs of %elements for the group <G> of orders <orderx> and <ordery>, where %<orderx> and <ordery> are positive integers. % %\>TranslatePresentation( <fgens>, <rels>, <sgens>, <newgens> ) F % %finds a new presentation, in terms of the generators <newgens>, for %the *simple* group $S = \langle <fgens> \mid <rels> \rangle$, with %subgroup generated by <sgens>; <fgens> should be a *pair* of free %group generators; <rels>, <sgens> and <newgens> should be lists of %words in <fgens>. Furthermore, <newgens> should be a list of two %words, the first of which should be of even order and the second of %odd prime order in the group $S$. The idea is that, by using Tietze %transformations, the variables `x' and `y' (see~"x") are assigned to %the words of <newgens>, which make up the new <fgens>, and <rels> and %<sgens> are written in terms of the new <fgens>, %`TranslatePresentation' returns a record with fields `fgens', `rels' %and `sgens' that are the new values of <fgens>, and <rels> and %<sgens>, respectively. % %\>IsACEResExampleOK() F % %does a number of integrity checks and tries to give an accurate %diagnosis if something is wrong, returning `true' if everything is %``OK'' and `false' otherwise. It is called whenever %`ACEReadResearchExample' is executed and is not intended for direct %usage by users. % %\>`ACEResExample' V % %is initially set to a record with one field `filename' whenever %`ACEReadResearchExample' is executed and the value set to that field %is the name of the file read by `ACEReadResearchExample'; other fields %are set as the need arises. Essentially, it is used as a temporary %variable store when `ACEReadResearchExample' reads a `do<XXX>.g' file. % %\>`ALLRELS' V %\>`newrels' V % %are initially set to null lists. Each time `PGRelFind' is executed %these each ``third'' relator tested is appended to `ALLRELS' and each %relator that is satisfied by the group defined by the <fgens> and %<rels> arguments of `PGRelFind' (see~"PGRelFind") is appended to %`newrels'. The user should normally reset these lists back to null %lists between invocations of `PGRelFind'. % %\>`F' V %\>`a' V %\>`b' V % %are the free group `F' and its two generators `a' and `b', used in the %presentations described below. % %\>`newF' V %\>`x' V %\>`y' V % %are the free group `newF' and its two generators `x' and `y'; the %record field `fgens' returned by `TranslatePresentation' is the list %`[x, y]' (see~"TranslatePresentation"). % %\>`L2_8' V %\>`L2_16' V %\>`L3_3s' V %\>`U3_3s' V %\>`M11' V %\>`M12' V %\>`L2_32' V %\>`U3_4s' V %\>`J1s' V %\>`L3_5s' V %\>`PSp4_4s' V % %contain presentations for the perfect simple groups $L_2(8)$, %$L_2(16)$, $L_3(3)$, $U_3(3)$, $M_{11}$, $M_{12}$, $L_2(32)$, $U_3(4)$, %$J_1$, $L_3(5)$ and $PSp_4(4)$, respectively. Those with names ending %in `s'contain lists of records (the `s' is meant to suggest ``plural''), %and those with no `s' are just single records, where each record has the %following fields: % %\beginitems % %\quad`source' & %a citation giving the source from which the presentation was obtained, %where `CCN85', `CMY79' and `CR84' indicate \cite{CCN85}, \cite{CMY79} %and \cite{CR84}, respectively, and `Bray' indicates the website %\URL{http://www.cix.co.uk/~vicarage/}; % %\quad`rels' & %the relators of the group in terms of the generators `a' and `b' %defined above (see~"a"); and % %\quad`sgens' & %the generators of a large (usually maximal) subgroup of the group, in %terms of the generators `a' and `b'. % %\enditems % %\>`presentations' V % %is a record whose fields are the names of the presentation variables %of~"M11", and whose values are the variables, %e.g.~`presentations.L2_16' is the same object as `L2_16'. % %*The `do<XXX>.g' files* % %Each `do<XXX>.g' file (other than the generic `doGrp.g') had a name %that is formed by taking one of the fields of `presentations', %removing any underscore or `s', prepending `"do"' and appending %`".g"'; also, each requires that you run % %\beginexample %gap> ACEReadResearchExample(); %\endexample % %first to define the functions described above. If you forget, never %mind~\dots~you will be reminded. % %Many of the outputs are fairly lengthy and you may wish to use `LogTo' %(see~"ref:LogTo") in order to peruse the output at leisure, later. %Also, for many of the larger groups one needs to be able to set %`ACEworkspace' larger than `10^6', but how large? The investigation %showing that $L_3(5)$ has deficiency 0 was run on an SGI Origin 2000 %with 6.4 Gb of RAM and with `ACEworkspace' set to `10^8', to find a %``third'' relator of 23 bisyllables. It turns out however that one %need only set `ACEworkspace := 5 * 10^6' to find that %relator~\dots~one can be very knowledgeable in hindsight! In the list %below we give values for `ACEworkspace' that will succeed in finding %the shortest ``third'' relator that we found, when the method %succeeded. (Of course, `PGRelFind' runs a lot faster if the workspace %given to {\ACE} is kept small.) % %Recall `ACEReadResearchExample' expects a filename (i.e.~a string), or %no argument; here are the *filenames* of the `do<XXX>.g' provided: % %\beginitems % %\quad`"doGrp.g"' & %This allows the user to experiment with any of the groups that have %data in the `presentations' record. The user *must* supply a value for %the following option: % %\qquad`grp := <presField>' & %<presField> must be the name (i.e.~a string) of a field in the %`presentations' record (this selects the group to be investigated); %and, possibly one or both of the following options: % %\qquad`n := <n>' & %If <presField>, above, ends in `s' then the user *must* supply a %positive integer <n> for `n', which indicates that the <n>th record of %`presentations.(<presField>)' is desired. % %\qquad`newgens := <wordlist>' & %<wordlist> must be a list of two words in the generators `a' and `b' %which are to be the words used for the <newgens> argument of %`TranslatePresentation' (see~"TranslatePresentation" for the %conditions they must satisfy). If `newgens' is not supplied, it is %assumed that `TranslatePresentation' is not needed, i.e.~that the %generators `a' and `b' are already an involution and an element of odd %prime order, respectively, and that `PGRelFind' may be immediately %applied. % %& %The user may also supply `PGRelFind' options, though unlike `grp' and %`newgens', they will not be directly observable in the output. %An example using both sets of options is given below. % %\quad`"doL28.g"' & %This provides a nice small example. You may supply any `PGRelFind' %options you wish, and/or you may use the following option which has %been specifically designed to help you get acquainted with the %`PGRelFind' options (only the equivalent of the `optex' options will %be explicitly observable in the output). % %\qquad`optex := <val>' & %<val> may be an integer or a list of integers in the range %$1,\ldots,8$. The output of `ACEReadResearchExample();' suggests a %combination that is 65 lines long. Most of the possible choices for %`optex' give an output of the order of 100 to 200 lines, which isn't %too bad if you have say an `xterm' window with the capacity to scroll %back. % %\quad`"doL216.g"' & %As is the case with the remaining `do<XXX>.g' files, there are no %options other than those for `PGRelFind'. It is sufficient to set %`ACEworkspace := 2 * 10^4'. % %\quad`"doL33.g"' & %It is sufficient to set `ACEworkspace := 2 * 10^4'. % %\quad`"doU33.g"' & %Though $U_3(3)$ is known to have deficiency 0, the %`TranslatePresentation'-`PGRelFind' method does not appear to be able %to demonstrate it. % %\quad`"doM11.g"' & %`ACEworkspace' needs to be set to at least `6 * 10^6'. % %\quad`"doL232.g"' & %It is sufficient to set `ACEworkspace := 5 * 10^4'. % %\quad`"doU34.g"' & %It is sufficient to set `ACEworkspace := 5 * 10^4'. % %\quad`"doJ1.g"' & %`ACEworkspace' needs to be set to at least `5 * 10^6'. % %\quad`"doL35.g"' & %`ACEworkspace' needs to be set to at least `5 * 10^6'. % %\quad`"doPSp44.g"' & %We were unable to determine whether $PSp_4(4)$ has deficiency 0, via %the `TranslatePresentation'-`PGRelFind' method. % %\enditems % %The following example shows that you can use both the options peculiar %to the file `doGrp.g' and the options of `PGRelFind' when %`ACEReadResearchExample' is called with argument (filename) %`"doGrp.g"'. % %\beginexample %gap> ACEReadResearchExample("doGrp.g" %> : grp := "L2_8", newgens := [a^3*b, a^2*b], %> head := x*y*x*y*x*y^-1*x*y*x*y); %# IsACEResExampleOK() sets ACEResExample.grp from options grp, n %# ACEResExample.newgens from option newgens %gap> ACEResExample.G := TranslatePresentation([a, b], %> ACEResExample.grp.rels, %> ACEResExample.grp.sgens, %> ACEResExample.newgens); %#I there are 4 generators and 6 relators of total length 45 %#I there are 2 generators and 4 relators of total length 55 %rec( % fgens := [ x, y ], % rels := [ x^2, y^3, y^-1*x*y^-1*x*y^-1*x*y^-1*x*y^-1*x*y^-1*x*y^-1*x, % x*y^-1*x*y*x*y^-1*x*y*x*y^-1*x*y*x*y^-1*x*y*x*y*x*y^-1*x*y*x*y^ % -1*x*y*x*y^-1*x*y*x*y^-1*x*y*x*y ], % sgens := [ x*y^-1 ] ) %gap> ACEResExample.Gn := PGRelFind(ACEResExample.G.fgens, %> ACEResExample.G.rels, %> ACEResExample.G.sgens); %GroupOrder=504 SubgroupIndex=72 %#bisyllables in head = 5 head: x*y*x*y*x*y^-1*x*y*x*y %Max #bisyllables in tail = 10 (granularity = 10) %#bisyllables in middle = 0 %Candidate relator: x*y*x*y*x*y^-1*x*y*x*y*x*y^-1*x*y^-1*x*y*x*y*x*y*x*y^ %-1*x*y^-1*x*y^-1*x*y*x*y^-1 % #bisyllables = 15 (#bisyllables in tail = 10) #words tested: 1 %ACEStats: % index=72 cputime=10 ms maxcosets=98 totcosets=121 %Large subgroup index OK %ACEStats for cyclic subgroup: % index=168 cputime=10 ms maxcosets=170 totcosets=232 %Cyclic subgroup index OK %Success! ... new relator: % x*y*x*y*x*y^-1*x*y*x*y*x*y^-1*x*y^-1*x*y*x*y*x*y*x*y^-1*x*y^-1*x*y^ %-1*x*y*x*y^-1 %... continuing (there may be a shorter relator). %Candidate relator: x*y*x*y*x*y^-1*x*y*x*y*x*y^-1*x*y*x*y^-1*x*y^-1*x*y^ %-1*x*y*x*y*x*y*x*y^-1*x*y^-1 % #bisyllables = 15 (#bisyllables in tail = 10) #words tested: 1 %ACEStats: % index=72 cputime=10 ms maxcosets=102 totcosets=123 %Large subgroup index OK %ACEStats for cyclic subgroup: % index=168 cputime=0 ms maxcosets=170 totcosets=236 %Cyclic subgroup index OK %Success! ... new relator: % x*y*x*y*x*y^-1*x*y*x*y*x*y^-1*x*y*x*y^-1*x*y^-1*x*y^-1*x*y*x*y*x*y*x*y^ %-1*x*y^-1 %... continuing (there may be a shorter relator). %Middles of length 0 exhausted. %Relator found of length 15, is shortest. %rec( % gens := [ x, y ], % rels := % [ x^2, y^3, x*y*x*y*x*y^-1*x*y*x*y*x*y^-1*x*y*x*y^-1*x*y^-1*x*y^-1*x*y*x*y*\ %x*y*x*y^-1*x*y^-1 ], % sgens := [ x*y^-1 ] ) %gap> %\endexample % %The following % %\beginexample %gap> ACEReadResearchExample("doGrp.g" : grp := "L3_3s", n := 1); %\endexample % %is essentially equivalent to % %\beginexample %gap> ACEReadResearchExample("doL33.g"); %\endexample % %Observe that the option `n' is needed because `grp' selects a list of %records (as indicated by the trailing `s' of `"L3_3s"'). Also, observe %that the `newgens' option was not used since we already have a %presentation where the generators `a' and `b' are an involution and of %odd prime order, respectively. %enddisplay
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