[SONATA-tutorial] 1 Groups

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# 1 Groups

### Sections

1. Thomas' and Wood's catalogue of small groups
2. Subgroups
3. Group endomorphisms
4. Finding a set of generators

SONATA adds some functions for groups. To use the functions provided by SONATA, one has to load it into GAP:

        gap> LoadPackage( "sonata" );

## 1.1 Thomas' and Wood's catalogue of small groups

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.

## 1.2 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.

        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 ]) ]

## 1.3 Group endomorphisms

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) ] -> [ (), () ] ]

## 1.4 Finding a set of generators

It is well known that for any finite p-group G the factor G/Φ(G) modulo the Frattini subgroup Φ(G) has order pδ(G), where δ(G) is the minimal number of generators of G. Moreover the representatives of the residue classes modulo Φ(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.

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