%
\Chapter{Groups}
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SONATA adds some functions for groups. To use the functions provided by
SONATA, one has to load it into GAP:
\beginexample
    gap> LoadPackage( "sonata" );
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\Section{Thomas' and Wood's catalogue of small groups}


        Most of the nonabelian groups (even small ones) do not have a
        popular name (as $S_3$ or $A_4$). 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'. 
\beginexample
    gap> G := DihedralGroup( 8 );     
    <pc group of size 8 with 3 generators>
    gap> IdTWGroup( G );
    [ 8, 4 ]
\endexample
        If we want to refer to the group with the name `8/4' directly we
        say
\beginexample
    gap> H := TWGroup( 8, 4 );
    8/4
\endexample
        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.
\beginexample
    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) ]
\endexample
        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. 
\beginexample
    gap> IsomorphismGroups(G,H);
    [ f1, f2, f3 ] -> [ (2,4), (1,2,3,4), (1,3)(2,4) ]
\endexample
        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'.
\beginexample
    gap> NumberSmallGroups( 32 );
    51
    gap> GTW32_46;
    32/46
    gap> GTW32_46 = TWGroup( 32, 46 );
    true
\endexample
        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.
\beginexample
    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 ]
\endexample
        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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\Section{Subgroups}
    

        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.
\beginexample
    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 ]) ]
\endexample
    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\Section{Group endomorphisms}


        Everybody knows that every automorphism of the symmetric group $S_3$
        (= `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 $S_3$ are inner automorphisms.
\beginexample
    gap> G := GTW6_2;
    6/2
    gap> Automorphisms( G );
    [ [ (1,2), (1,2,3) ] -> [ (2,3), (1,2,3) ], 
      [ (2,3), (1,3,2) ] -> [ (1,2), (1,2,3) ], 
      [ (2,3), (1,3,2) ] -> [ (1,3), (1,2,3) ], 
      [ (2,3), (1,3,2) ] -> [ (2,3), (1,3,2) ], 
      [ (2,3), (1,3,2) ] -> [ (1,2), (1,3,2) ], 
      [ (2,3), (1,3,2) ] -> [ (2,3), (1,2,3) ] ]
    gap> Endos := Endomorphisms( G );
    [ [ (2,3), (1,3,2) ] -> [ (), () ], [ (2,3), (1,3,2) ] -> [ (2,3), () ],
      [ (2,3), (1,3,2) ] -> [ (1,2), () ], 
      [ (2,3), (1,3,2) ] -> [ (1,3), () ], 
      [ (1,2), (1,2,3) ] -> [ (2,3), (1,2,3) ], 
      [ (2,3), (1,3,2) ] -> [ (1,2), (1,2,3) ], 
      [ (2,3), (1,3,2) ] -> [ (1,3), (1,2,3) ], 
      [ (2,3), (1,3,2) ] -> [ (2,3), (1,3,2) ], 
      [ (2,3), (1,3,2) ] -> [ (1,2), (1,3,2) ], 
      [ (2,3), (1,3,2) ] -> [ (2,3), (1,2,3) ] ]
\endexample
        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*).
\beginexample
    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
\endexample
        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 $S_3$, 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.
\beginexample
    gap> e := Endos[4];
    [ (2,3), (1,3,2) ] -> [ (1,3), () ]
    gap> IsFixedpointfree( e );
    false
\endexample
        Now we filter out the fixed-point-free endomorphisms.
\beginexample
    gap> Filtered( Endos, IsFixedpointfree );
    [ [ (2,3), (1,3,2) ] -> [ (), () ] ]
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\Section{Finding a set of generators}


        It is well known that for any finite p-group $G$ the factor $G/\Phi(G)$
        modulo the Frattini subgroup $\Phi(G)$ has order $p^{\delta(G)}$, where
        $\delta(G)$ is the minimal number of generators of $G$. Moreover
        the representatives of the residue classes modulo $\Phi(G)$ 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).
\beginexample
    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, 2, 4, 7,11,10,14,16)( 3, 6, 8,13,15, 5, 9,12), 
      ( 1, 3)( 2, 5)( 4, 8)( 6,10)( 7,12)( 9,14)(11,15)(13,16), 
      ( 1, 5,14,13,11, 6, 4,12)( 2, 8,16, 3,10, 9, 7,15) ]
    gap> H := Group( last );
    Group([ ( 1, 2, 4, 7,11,10,14,16)( 3, 6, 8,13,15, 5, 9,12), 
      ( 1, 3)( 2, 5)( 4, 8)( 6,10)( 7,12)( 9,14)(11,15)(13,16), 
      ( 1, 5,14,13,11, 6, 4,12)( 2, 8,16, 3,10, 9, 7,15) ])
    gap> G = H;  # test
    true
\endexample
        The variable `last' in the this example refers to the last result,
        i.e. in this case the list of representatives.

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