[FGA] 2 Functionality of the FGA package

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# 2 Functionality of the FGA package

### Sections

1. New operations for free groups
2. Method installations
3. Constructive membership test
4. Automorphism groups of free groups

This chapter describes methods available from the FGA package.

In the following, let f be a free group created by `FreeGroup(`n`)`, and let u, u1 and u2 be finitely generated subgroups of f created by `Group` or `Subgroup`, or computed from some other subgroup of f. Let elm be an element of f.

For example:

    gap> f := FreeGroup( 2 );
    <free group on the generators [ f1, f2 ]>
    gap> u := Group( f.1^2, f.2^2, f.1*f.2 );
    Group([ f1^2, f2^2, f1*f2 ])
    gap> u1 := Subgroup( u, [f.1^2, f.1^4*f.2^6] );
    Group([ f1^2, f1^4*f2^6 ])
    gap> elm := f.1;
    f1
    gap> u2 := Normalizer( u, elm );
    Group([ f1^2 ])

## 2.1 New operations for free groups

These new operations are available for finitely generated subgroups of free groups:

- `FreeGeneratorsOfGroup(` u`) A`

  returns a list of free generators of the finitely generated subgroup u of a free group.

  The elements in this list form an N-reduced set. In addition to being a free (and thus minimal) generating set for u, this means that whenever v1, v2 and v3 are elements or inverses of elements of this list, then

  - v1 v2 ≠ 1 implies |v1 v2 | ≥ max(|v1 |, |v2 |), and
  - v1 v2 ≠ 1 and v2 v3 ≠ 1 implies |v1 v2 v3 | > |v1 | − |v2 | + |v3 |

  hold, where |·| denotes the word length.

- `RankOfFreeGroup(` u`) A`
- `Rank(` u`) O`

  returns the rank of the finitely generated subgroup u of a free group.

- `CyclicallyReducedWord(` elm`) O`

  returns the cyclically reduced form of elm.

  ## 2.2 Method installations

  This section lists operations that are already known to GAP. FGA installs new methods for them so that they can also be used with free groups. In cases where FGA installs methods that are usually only used internally, user functions are shown instead.

- `Normalizer(` u1`,` u2`) O`
- `Normalizer(` u`,` elm`) O`

  The first variant returns the normalizer of the finitely generated subgroup u2 in u1.

  The second variant returns the normalizer of 〈elm 〉 in the finitely generated subgroup u (see Normalizer in the Reference Manual).

- `RepresentativeAction(` u`,` d`,` e`) O`
- `IsConjugate(` u`,` d`,` e`) O`

  `RepresentativeAction` returns an element r ∈ u , where u is a finitely generated subgroup of a free group, such that d r =e , or fail, if no such r exists. d and e may be elements or subgroups of u.

  `IsConjugate` returns a boolean indicating whether such an element r exists.

- `Centralizer(` u`,` u2`) O`
- `Centralizer(` u`,` elm`) O`

  returns the centralizer of u2 or elm in the finitely generated subgroup u of a free group.

- `Index(` u1`,` u2`) O`
- `IndexNC(` u1`,` u2`) O`

  return the index of u2 in u1, where u1 and u2 are finitely generated subgroups of a free group. The first variant returns fail if u2 is not a subgroup of u1, the second may return anything in this case.

- `Intersection(` u1`,` u2`...) F`

  returns the intersection of u1 and u2, where u1 and u2 are finitely generated subgroups of a free group.

- ``elm`in` u`O`

  tests whether elm is contained in the finitely generated subgroup u of a free group.

- `IsSubgroup(` u1`,` u2`) F`

  tests whether u2 is a subgroup of u1, where u1 and u2 are finitely generated subgroups of a free group.

- ``u1`=` u2`O`

  test whether the two finitely generated subgroups u1 and u2 of a free group are equal.

- `MinimalGeneratingSet(` u`) A`
- `SmallGeneratingSet(` u`) A`
- `GeneratorsOfGroup(` u`) A`

  return generating sets for the finitely generated subgroup u of a free group. `MinimalGeneratingSet` and `SmallGeneratingSet` return the same free generators as `FreeGeneratorsOfGroup`, which are in fact a minimal generating set. `GeneratorsOfGroup` also returns these generators, if no other generators were stored at creation time.

  ## 2.3 Constructive membership test

  It is not only possible to test whether an element is in a finitely generated subgroup of free group, this can also be done constructively. The idiomatic way to do so is by using a homomorphism.

  Here is an example that computes how to write `f.1^2` in the generators `a=f1^2*f2^2` and `b=f.1^2*f.2`, checks the result, and then tries to write `f.1` in the same generators:

      gap> f := FreeGroup( 2 );
      <free group on the generators [ f1, f2 ]>
      gap> ua := f.1^2*f.2^2;; ub := f.1^2*f.2;;
      gap> u := Group( ua, ub );;
      gap> g := FreeGroup( "a", "b" );;
      gap> hom := GroupHomomorphismByImages( g, u,
      >             GeneratorsOfGroup(g),
      >             GeneratorsOfGroup(u) );
      [ a, b ] -> [ f1^2*f2^2, f1^2*f2 ]
      gap> # how can f.1^2 be expressed?
      gap> PreImagesRepresentative( hom, f.1^2 );
      b*a^-1*b
      gap> last ^ hom; # check this
      f1^2
      gap> ub * ua^-1 * ub; # another check
      f1^2
      gap> PreImagesRepresentative( hom, f.1 ); # try f.1
      fail
      gap> f.1 in u;
      false

  There are also lower level operations to get the same results.

- `AsWordLetterRepInGenerators(` elm`,` u`) O`
- `AsWordLetterRepInFreeGenerators(` elm`,` u`) O`

  return a letter representation (see Section Representations for Associative Words in the GAP Reference Manual) of the given elm relative to the generators the group was created with or the free generators as returned by `FreeGeneratorsOfGroup`.

  Continuing the above example:

      gap> AsWordLetterRepInGenerators( f.1^2, u );
      [ 2, -1, 2 ]
      gap> AsWordLetterRepInFreeGenerators( f.1^2, u );
      [ 2 ]

  This means: to get `f.1^2`, multiply the second of the given generators with the inverse of the first and again with the second; or just take the second free generator.

  ## 2.4 Automorphism groups of free groups

  The FGA package knows presentations of the automorphism groups of free groups. It also allows to express an automorphism as word in the generators of these presentations. This sections repeats the GAP standard methods to do so and shows functions to obtain the generating automorphisms.

- `AutomorphismGroup(` u`) A`

  returns the automorphism group of the finitely generated subgroup u of a free group.

  Only a few methods will work with this group. But there is a way to obtain an isomorphic finitely presented group:

- `IsomorphismFpGroup(` group`) A`

  returns an isomorphism of group to a finitely presented group. For automorphism groups of free groups, the FGA package implements the presentations of Neumann33. The finitely presented group itself can then be obtained with the command `Range`.

  Here is an example:

      gap> f := FreeGroup( 2 );;
      gap> a := AutomorphismGroup( f );;
      gap> iso := IsomorphismFpGroup( a );;
      gap> Range( iso );
      <fp group on the generators [ O, P, U ]>

  To express an automorphism as word in the generators of the presentation, just apply the isomorphism obtained from `IsomorphismFpGroup`.

      gap> aut := GroupHomomorphismByImages( f, f,
      >              GeneratorsOfGroup( f ), [ f.1^f.2, f.1*f.2 ] );
      [ f1, f2 ] -> [ f2^-1*f1*f2, f1*f2 ]
      gap> ImageElm( iso, aut );
      O^2*U*O*P^-1*U

  It is also possible to use `aut^iso` or `Image( iso, aut )`. Using `Image` will perform additional checks on the arguments.

  The FGA package provides a simpler way to create endomorphisms:

- `FreeGroupEndomorphismByImages(` g`,` imgs`) F`

  returns the endomorphism that maps the free generators of the finitely generated subgroup g of a free group to the elements listed in imgs. You may then apply `IsBijective` to check whether it is an automorphism.

  The follwowing functions return automorphisms that correspond to the generators in the presentation:

- `FreeGroupAutomorphismsGeneratorO(` group`) F`
- `FreeGroupAutomorphismsGeneratorP(` group`) F`
- `FreeGroupAutomorphismsGeneratorU(` group`) F`
- `FreeGroupAutomorphismsGeneratorS(` group`) F`
- `FreeGroupAutomorphismsGeneratorT(` group`) F`
- `FreeGroupAutomorphismsGeneratorQ(` group`) F`
- `FreeGroupAutomorphismsGeneratorR(` group`) F`

  return the automorphism which maps the free generators [ x1, x2, ..., xn ] of group to

  O:
    [ x1−1, x2, ..., xn ] (n ≥ 1)
  P:
    [ x2, x1, x3, ..., xn ] (n ≥ 2)
  U:
    [ x1x2, x2, x3, ..., xn ] (n ≥ 2)
  S:
    [ x2−1, x3−1, ..., xn−1, x1−1 ] (n ≥ 1)
  T:
    [ x2, x1−1, x3, ..., xn ] (n ≥ 2)
  Q:
    [ x2, x3, ..., xn, x1 ] (n ≥ 2)
  R:
    [ x2−1, x1, x3, x4, ..., xn−2, xnxn−1−1, xn−1−1 ] (n ≥ 4)

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