SONATA adds some functions for groups. To use the functions provided by SONATA, one has to load it into GAP:
gap> LoadPackage( "sonata" );
Most of the nonabelian groups (even small ones) do not have a
popular name (as S3 or A4). We like to give unique names to
the groups we are working with. The book ``Group Tables'' by
Thomas and Wood classifies all groups up to order 32. In this book
every group has a name of the form m/n, where m is the order of
the group and n the number of the particular group of order m.
The cyclic groups have the name m/1. Then come the abelian groups,
finally the non-abelian ones. To find out the name of a given group
in their book we use IdTWGroup.
gap> G := DihedralGroup( 8 );
<pc group of size 8 with 3 generators>
gap> IdTWGroup( G );
[ 8, 4 ]
If we want to refer to the group with the name 8/4 directly we
say
gap> H := TWGroup( 8, 4 );
8/4
Groups which are obtained in this way always come as a group of
permutations. We can have a look at the elements of H if we ask
for H as a list.
gap> AsList( H );
[ (), (2,4), (1,2)(3,4), (1,2,3,4), (1,3), (1,3)(2,4), (1,4,3,2),
(1,4)(2,3) ]
Clearly, G and H are not equal but they are isomorphic. If we want
to know what the isomorphism between the two looks like, we use
IsomorphismGroups. Note, that a homomorphism is determined by the
images of the generators.
gap> IsomorphismGroups(G,H);
[ f1, f2, f3 ] -> [ (2,4), (1,2,3,4), (1,3)(2,4) ]
How many nonisomorphic groups are there of order n? Up to order
1000 the function NumberSmallGroups gives the answer. As a shortcut
for TWGroup( 32, 46 ) we may also type GTW32_46.
gap> NumberSmallGroups( 32 );
51
gap> GTW32_46;
32/46
gap> GTW32_46 = TWGroup( 32, 46 );
true
Now we find all nonabelian groups with trivial centre of order at most
32. We use GroupList, a list of all groups up to order 32 and filter
out the nonabelian ones with trivial center.
gap> Filtered( GroupList, g -> not IsAbelian( g ) and
> Size(Centre( g ))=1 );
[ 6/2, 10/2, 12/4, 14/2, 18/4, 18/5, 20/5, 21/2, 22/2, 24/12, 26/2,
30/4 ]
This was the first time that we have used a function as an argument.
The second argument of the function Filtered is a function
(g -> not ...), which returns for every g the boolean value true
if g is not abelian and the size of its centre is 1, and false
otherwise. This is the easiest way to write a function.
The function Subgroups returns a list of all subgroups of a group.
We can use this function and the Filtered command to determine all
characteristic subgroups of the dihedral group of order 16.
gap> D16 := DihedralGroup( 16 );
<pc group of size 16 with 4 generators>
gap> S := Subgroups( D16 );
[ Group([ ]), Group([ f4 ]), Group([ f1 ]), Group([ f1*f3 ]),
Group([ f1*f4 ]), Group([ f1*f3*f4 ]), Group([ f1*f2 ]),
Group([ f1*f2*f3 ]), Group([ f1*f2*f4 ]),
Group([ f1*f2*f3*f4 ]), Group([ f4, f3 ]), Group([ f4, f1 ]),
Group([ f1*f3, f4 ]), Group([ f4, f1*f2 ]),
Group([ f1*f2*f3, f4 ]), Group([ f4, f3, f1 ]),
Group([ f4, f3, f2 ]), Group([ f4, f3, f1*f2 ]),
Group([ f4, f3, f1, f2 ]) ]
gap> C := Filtered( S, G -> IsCharacteristicInParent( G ) );
[ Group([ ]), Group([ f4 ]), Group([ f4, f3 ]), Group([ f4, f3, f2 ]),
Group([ f4, f3, f1, f2 ]) ]
Everybody knows that every automorphism of the symmetric group S3
(= GTW6_2) fixes a point (besides the identity of the group). But,
are there endomorphisms which fix nothing but the identity? We are
going to simply try it out. On our way we will find out that all
automorphisms of S3 are inner automorphisms.
gap> G := GTW6_2;
6/2
gap> Automorphisms( G );
[ IdentityMapping( 6/2 ), ^(2,3), ^(1,3), ^(1,3,2), ^(1,2,3), ^(1,2) ]
gap> Endos := Endomorphisms( G );
[ [ (1,2), (1,2,3) ] -> [ (), () ], [ (1,2), (1,2,3) ] -> [ (2,3), () ],
[ (1,2), (1,2,3) ] -> [ (1,3), () ], [ (1,2), (1,2,3) ] -> [ (1,2), () ],
[ (1,2), (1,2,3) ] -> [ (2,3), (1,2,3) ],
[ (1,2), (1,2,3) ] -> [ (2,3), (1,3,2) ],
[ (1,2), (1,2,3) ] -> [ (1,2), (1,3,2) ],
[ (1,2), (1,2,3) ] -> [ (1,2), (1,2,3) ],
[ (1,2), (1,2,3) ] -> [ (1,3), (1,2,3) ],
[ (1,2), (1,2,3) ] -> [ (1,3), (1,3,2) ] ]
Now it is time for real programming, but don't worry, it is all very
simple. We write a function which decides whether an endomorphism
fixes a point besides the identity or not (in the latter case we
call the endomorphism fixed-point-free).
gap> IsFixedpointfree := function( endo )
>local group;
> group := Source( endo ); # the domain of endo
> return ForAll( group, x -> (x <> x^endo) or (x = Identity(group)) );
> # x is not fixed or x is the identity
>end;
function ( endo ) ... end
This paragraph says that IsFixedpointfree is a function that takes
one argument (called endo). Now we create a local variable group to
store the group on which the endomorphism acts (in our example this
will always be S3, but maybe we want to use this function for
other groups, too). Local means that GAP may forget this variable
as soon as it has computed what we want (and it will forget it
instantly afterwards). Now we store the domain of endo in the
variable group. The next line already returns the result. It returns
true if for all elements x of group either x is not fixed
by endo or x is the identity of the group. This line is a
one-to-one translation of the logical statement that endo is
fixed-point-free.
The result is a function which can be applied to any endomorphism, now.
For example we can ask if the fourth endomorphism in the list E is
fixed-point-free.
gap> e := Endos[4];
[ (1,2), (1,2,3) ] -> [ (1,2), () ]
gap> IsFixedpointfree( e );
false
Now we filter out the fixed-point-free endomorphisms.
gap> Filtered( Endos, IsFixedpointfree );
[ [ (1,2), (1,2,3) ] -> [ (), () ] ]
It is well known that for any finite p-group G the factor G/F(G) modulo the Frattini subgroup F(G) has order pd(G), where d(G) is the minimal number of generators of G. Moreover the representatives of the residue classes modulo F(G) form a set of generators. So a generating set for a p-group could be obtained in the following way. We choose the group 16/11 (a semidirect product of the cyclic group of order 8 with the cyclic group of order 2).
gap> G := GTW16_11;
16/11
gap> F := FrattiniSubgroup( G );
Group([ (1,4,11,14)(2,7,10,16)(3,8,15,9)(5,12,6,13) ])
gap> NontrivialRepresentativesModNormalSubgroup( G, F );
[ (1,16,14,10,11,7,4,2)(3,12,9,5,15,13,8,6),
(1,3)(2,5)(4,8)(6,10)(7,12)(9,14)(11,15)(13,16),
(1,13,4,5,11,12,14,6)(2,3,7,8,10,15,16,9) ]
gap> H := Group( last );
Group([ (1,16,14,10,11,7,4,2)(3,12,9,5,15,13,8,6),
(1,3)(2,5)(4,8)(6,10)(7,12)(9,14)(11,15)(13,16),
(1,13,4,5,11,12,14,6)(2,3,7,8,10,15,16,9) ])
gap> G = H; # test
true
The variable last in the this example refers to the last result,
i.e. in this case the list of representatives.
SONATA-tutorial manual