When you start GAP it already knows several groups. For example, some basic groups such as cyclic groups or symmetric groups, all primitive permutation groups of degree at most 50, and all 2-groups of size at most 256.
Each of the sets above is called a group library. The set of all groups that GAP knows initially is called the collection of group libraries.
In this section we show you how you can access the groups in those libraries and how you can extract groups with certain properties from those libraries.
Let us start with the basic groups, because they are not accessed in the same way as the groups in the other libraries.
To access such a basic group you just call a function with an appropriate
name, such as CyclicGroup or SymmetricGroup.
gap> c13 := CyclicGroup( 13 );
Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13) )
gap> Size( c13 );
13
gap> s8 := SymmetricGroup( 8 );
Group( (1,8), (2,8), (3,8), (4,8), (5,8), (6,8), (7,8) )
gap> Size( s8 );
40320
The functions above also accept an optional first argument that describes
the type of group. For example you can pass AgWords to CyclicGroup
Finite Polycyclic Groups).
gap> c13 := CyclicGroup( AgWords, 13 );
Group( c13 )
Of course you cannot pass AgWords to SymmetricGroup, because
symmetric groups are in general not polycyclic.
The default is to construct the groups as permutation groups, but you can
also explicitly pass Permutations. Other possible arguments are
AgWords for finite polycyclic groups, Words for finitely presented
groups, and Matrices for matrix groups (however only Permutations and
AgWords currently work).
Let us now turn to the other libraries. They are all accessed in a uniform way. For a first example we will use the group library of primitive permutation groups.
To extract a group from a group library you generally use the extraction
function. In our example this function is called PrimitiveGroup. It
takes two arguments. The first is the degree of the primitive
permutation group that you want and the second is an integer that
specifies which of the primitive permutation groups of that degree you
want.
gap> g := PrimitiveGroup( 12, 3 );
M(11)
gap> g.generators;
[ ( 2, 6)( 3, 5)( 4, 7)( 9,10), ( 1, 5, 7)( 2, 9, 4)( 3, 8,10),
( 1,11)( 2, 7)( 3, 5)( 4, 6), ( 2, 5)( 3, 6)( 4, 7)(11,12) ]
gap> Size( g );
7920
gap> IsSimple( g );
true
gap> h := PrimitiveGroup( 16, 19 );
2^4.A(7)
gap> Size( h );
40320
The reason for the extraction function is as follows. A group library is usually not stored as a list of groups. Instead a more compact representation for the groups is used. For example the groups in the library of 2-groups are represented by 4 integers. The extraction function hides this representation from you, and allows you to access the group library as if it was a table of groups (two dimensional in the above example).
What arguments the extraction function accepts, and how they are interpreted is described in the sections that describe the individual group libraries in chapter Group Libraries. Those functions will of course signal an error when you pass illegal arguments.
Suppose that you want to get a list of all primitive permutation groups
that have a degree 10 and are simple but not cyclic. It would be very
difficult to use the extraction function to extract all groups in the
group library, and test each of those. It is much simpler to use the
selection function. The name of the selection function always begins
with All and ends with Groups, in our example it is thus called
AllPrimitiveGroups.
gap> AllPrimitiveGroups( DegreeOperation, 10,
> IsSimple, true,
> IsCyclic, false );
[ A(5), PSL(2,9), A(10) ]
AllPrimitiveGroups takes a variable number of argument pairs consisting
of a function (e.g. DegreeOperation) and a value (e.g. 10). To
understand what AllPrimitiveGroups does, imagine that the group library
was stored as a long list of permutation groups. AllPrimitiveGroups
takes all those groups in turn. To each group it applies each function
argument and compares the result with the corresponding value argument.
It selects a group if and only if all the function results are equal to
the corresponding value. So in our example AllPrimitiveGroups selects
those groups g for which DegreeOperation(g) = 10 and
IsSimple(g) = true and IsCyclic(g) = false. Finally
AllPrimitiveGroups returns the list of the selected groups.
Next suppose that you want all the primitive permutation groups that have
degree at most 10, are simple but are not cyclic. You could obtain
such a list with 10 calls to AllPrimitiveGroups (i.e., one call for the
degree 1 groups, another for the degree 2 groups and so on), but there is
a simple way. Instead of specifying a single value that a function must
return you can simply specify a list of such values.
gap> AllPrimitiveGroups( DegreeOperation, [1..10],
> IsSimple, true,
> IsCyclic, false );
[ A(5), PSL(2,5), A(6), PSL(3,2), A(7), PSL(2,7), A(8), PSL(2,8),
A(9), A(5), PSL(2,9), A(10) ]
Note that the list that you get contains A(5) twice, first in its
primitive presentation on 5 points and second in its primitive
presentation on 10 points.
Thus giving several argument pairs to the selection function allows you to express the logical and of properties that a group must have to be selected, and giving a list of values allows you to express a (restricted) logical or of properties that a group must have to be selected.
There is no restriction on the functions that you can use. It is even possible to use functions that you have written yourself. Of course, the functions must be unary, i.e., accept only one argument, and must be able to deal with the groups.
gap> NumberConjugacyClasses := function ( g )
> return Length( ConjugacyClasses( g ) );
> end;
function ( g ) ... end
gap> AllPrimitiveGroups( DegreeOperation, [1..10],
> IsSimple, true,
> IsCyclic, false,
> NumberConjugacyClasses, 9 );
[ A(7), PSL(2,8) ]
Note that in some cases a selection function will issue a warning. For
example if you call AllPrimitiveGroups without specifying the degree,
it will issue such a warning.
gap> AllPrimitiveGroups( Size, [100..400],
> IsSimple, true,
> IsCyclic, false );
#W AllPrimitiveGroups: degree automatically restricted to [1..50]
[ A(6), PSL(3,2), PSL(2,7), PSL(2,9), A(6) ]
If selection functions would really run over the list of all groups in a group library and apply the function arguments to each of those, they would be very inefficient. For example the 2-groups library contains 58760 groups. Simply creating all those groups would take a very long time.
Instead selection functions recognize certain functions and handle them
more efficiently. For example AllPrimitiveGroups recognizes
DegreeOperation. If you pass DegreeOperation to AllPrimitiveGroups
it does not create a group to apply DegreeOperation to it. Instead it
simply consults an index and quickly eliminates all groups that have a
different degree. Other functions recognized by AllPrimitiveGroups are
IsSimple, Size, and Transitivity.
So in our examples AllPrimitiveGroups, recognizing DegreeOperation
and IsSimple, eliminates all but 16 groups. Then it creates those 16
groups and applies IsCyclic to them. This eliminates 4 more groups
(C(2), C(3), C(5), and C(7)). Then in our last example it
applies NumberConjugacyClasses to the remaining 12 groups and
eliminates all but A(7) and PSL(2,8).
The catch is that the selection functions will take a large amount of
time if they cannot recognize any special functions. For example the
following selection will take a large amount of time, because only
IsSimple is recognized, and there are 116 simple groups in the
primitive groups library.
AllPrimitiveGroups( IsSimple, true, NumberConjugacyClasses, 9 );
So you should specify a sufficiently large set of recognizable functions when you call a selection function. It is also advisable to put those functions first (though in some group libraries the selection function will automatically rearrange the argument pairs so that the recognized functions come first). The sections describing the individual group libraries in chapter Group Libraries tell you which functions are recognized by the selection function of that group library.
There is another function, called the example function that behaves
similar to the selection function. Instead of returning a list of all
groups with a certain set of properties it only returns one such group.
The name of the example function is obtained by replacing All by One
and stripping the s at the end of the name of the selection function.
gap> OnePrimitiveGroup( DegreeOperation, [1..10],
> IsSimple, true,
> IsCyclic, false,
> NumberConjugacyClasses, 9 );
A(7)
The example function works just like the selection function. That means that all the above comments about the special functions that are recognized also apply to the example function.
Let us now look at the 2-groups library. It is accessed in the same way
as the primitive groups library. There is an extraction function
TwoGroup, a selection function AllTwoGroups, and an example function
OneTwoGroup.
gap> g := TwoGroup( 128, 5 );
Group( a1, a2, a3, a4, a5, a6, a7 )
gap> Size( g );
128
gap> NumberConjugacyClasses( g );
80
The groups are all displayed as Group( a1, a2, ..., an ), where 2^n
is the size of the group.
gap> AllTwoGroups( Size, 256,
> Rank, 3,
> pClass, 2 );
[ Group( a1, a2, a3, a4, a5, a6, a7, a8 ),
Group( a1, a2, a3, a4, a5, a6, a7, a8 ),
Group( a1, a2, a3, a4, a5, a6, a7, a8 ),
Group( a1, a2, a3, a4, a5, a6, a7, a8 ) ]
gap> l := AllTwoGroups( Size, 256,
> Rank, 3,
> pClass, 5,
> g -> Length( DerivedSeries( g ) ), 4 );;
gap> Length( l );
28
The selection and example function of the 2-groups library recognize
Size, Rank, and pClass. Note that Rank and pClass are
functions that can in fact only be used in this context, i.e., they can
not be applied to arbitrary groups.
The following discussion is a bit technical and you can ignore it safely.
For very big group libraries, such as the 2-groups library, the groups (or their compact representations) are not stored on a single file. This is because this file would be very large and loading it would take a long time and a lot of main memory.
Instead the groups are stored on a small number of files (27 in the case of the 2-groups). The selection and example functions are careful to load only those files that may actually contain groups with the specified properties. For example in the above example the files containing the groups of size less than 256 are never loaded. In fact in the above example only one very small file is loaded.
When a file is loaded the selection and example functions also unload the previously loaded file. That means that they forget all the groups in this file again (except those selected of course). Thus even if the selection or example functions have to search through the whole group library, only a small part of the library is held in main memory at any time. In principle it should be possible to search the whole 2-groups library with as little as 2 MByte of main memory.
If you have sufficient main memory available you can explicitly load
files from the 2-groups library with ReadTwo( filename ), e.g.,
Read( "twogp64" ) to load the file with the groups of size 64. Those
files will then not be unloaded again. This will take up more main
memory, but the selection and example function will work faster, because
they do not have to load those files again each time they are needed.
In this section you have seen the basic groups library and the group libraries of primitive groups and 2-groups. You have seen how you can extract a single group from such a library with the extraction function. You have seen how you can select groups with certain properties with the selection and example function. Chapter Group Libraries tells you which other group libraries are available.
GAP 3.4.4