[SONATA] 9 Fixed-point-free automorphism groups

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# 9 Fixed-point-free automorphism groups

### Sections

1. Fixed-point-free automorphism groups and Frobenius groups
2. Fixed-point-free representations
3. Fixed-point-free automorphism groups

The functions described in this chapter are purely group-theoretic and are meant to provide solvable fixed-point-free automorphism groups acting on abelian groups (Frobenius groups with abelian Frobenius kernel and solvable Frobenius complement) for the construction of centralizer nearrings, planar nearrings, designs, and so on.

The classification of fixed-point-free automorphism groups in types I - IV follows Zassenhaus' papers and [Wolf:Spaces]. The fixed-point-free automorphism groups acting on abelian groups are constructed from fixed-point-free representations as described in [Mayr:Representations].

## 9.1 Fixed-point-free automorphism groups and Frobenius groups

- `IsFpfAutomorphismGroup(` phi`,` G`)`

  An automorphism group F of a group G acts fixed-point-free (fpf) on G if F has more than 1 element and no automorphism in F except the identity mapping has a fixed point besides the group identity of G.

  F is fpf on G, iff the semidirect product of G and F, with F acting naturally on G, is a Frobenius group.

  The function `IsFpfAutomorphismGroup` returns the according value `true` or `false` for a group of automorphisms phi on the group G.

          gap> C9 := CyclicGroup( 9 );
          <pc group of size 9 with 2 generators>
          gap> a := GroupHomomorphismByFunction( C9, C9, x -> x^-1 );;
          gap> phi := Group( a );;
          gap> Size( phi );
          2
          gap> IsFpfAutomorphismGroup( phi, C9 );
          true

- `FpfAutomorphismGroupsMaxSize(` G`)`

  `FpfAutomorphismGroupsMaxSize` returns a list with integers kmax and dmax where kmax is an upper bound for the size of an fpf automorphism group on the group G; for example, the order of G is congruent to 1 modulo kmax and kmax is odd for nonabelian groups G. The order of any fpf automorphism group phi on G divides kmax.

  Let phi be a metacyclic fpf automorphism group acting on G. Then phi has a cyclic normal subgroup whose index in phi divides dmax. Thus, if dmax is 1, then G admits cyclic fpf automorphism groups only.

          gap> G := ElementaryAbelianGroup( 49 );;
          gap> FpfAutomorphismGroupsMaxSize( G );
          [ 48, 2 ]
          gap> C15 := CyclicGroup( 15 );;
          gap> FpfAutomorphismGroupsMaxSize( C15 );
          [ 2, 1 ]

- `FrobeniusGroup(` phi`,` G`)`

  `FrobeniusGroup` constructs the semidirect product of G with the fpf automorphism group phi of G with the multiplication (a,g)*(b,h) = (ab,gah) by using the function `SemidirectProduct`. Thus a Frobenius group with Frobenius kernel G and Frobenius complement phi where the action of phi on G is the natural action of automorphisms on the group is returned.

  The unique Frobenius group with kernel G = (Z3)2×(Z5)2 and quaternion complement is constructed as follows:

          gap> aux := FpfAutomorphismGroupsMetacyclic( [3,3,5,5], 4, -1 );
          [ [ [ [ f1, f2, f3, f4 ] -> [ f1^2, f2^3, f3*f4, f3*f4^2 ],
                    [ f1, f2, f3, f4 ] -> [ f2^4, f1, f4^2, f3 ] ] ],
            <pc group of size 225 with 4 generators> ]
          gap> phi := Group( aux[1][1] );
          <group with 2 generators>
          gap> G := aux[2];
          <pc group of size 225 with 4 generators>
          gap> FrobeniusGroup( phi, G );
          <pc group of size 1800 with 7 generators>

  ## 9.2 Fixed-point-free representations

- `IsFpfRepresentation(` matrices`,` F`)`

  Let p be a representation of the group F over the finite field F. If for all j Î F except for the identity the matrix p(j) does not have 1 as an eigenvalue, then p is said to be fpf.

  Let p be an fpf representation of F over F with degree d. Then p is faithful, the order of F and the characteristic of F are coprime and p is a sum of irreducible faithful fpf F-representations. The matrix group p(F) acts fpf on the vectorspace Fd.

  For a list of d×d matrices, matrices, over the field F, the function `IsFpfRepresentation` returns `true` if the group generated by matrices acts fpf on the d-dimensional vectorspace over F, and `false` otherwise.

          gap> F := GF(5);;
          gap> A := [[2,0],[0,3]]*One(F);
          [ [ Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^3 ] ]
          gap> IsFpfRepresentation( [A], F );
          true

- `DegreeOfIrredFpfRepCyclic(` p`,` m`)`

  returns the degree of the irreducible fpf representations of the cyclic group of order m over GF(p), where m and p are coprime.

  Note, that all irreducible fpf representations of the cyclic group of order m over GF(p) have the same degree, the multiplicative order of p modulo m, `OrderMod( p, m )`.

          gap> DegreeOfIrredFpfRepCyclic( 5, 9 );
          6

- `DegreeOfIrredFpfRepMetacyclic(` p`,` m`,` r`)`

  returns the degree of the irreducible fpf representations of the metacyclic group F determined by parameters m and r over GF(p). If the parameters are not feasible, then an error is returned. See `FpfRepresentationsMetacyclic` for a presentation of this group.

  All irreducible fpf representations of the metacyclic group over GF(p) have the same degree, namely the size of multiplicative group generated by p and r modulo m.

  We determine the degree of the irreducible fpf representation of the quaternion group over GF(5):

          gap> DegreeOfIrredFpfRepMetacyclic( 5, 4, -1 );
          2

- `DegreeOfIrredFpfRep2(` p`,` m`,` r`,` k`)`

  returns the degree of the irreducible fpf representations of the type-II-group F determined by parameters m, r, and k over GF(p). If the parameters are not feasible or if the parameters describe the presentation of a metacyclic group, then an error is returned. See `FpfRepresentations2` for a presentation of this group.

  All irreducible fpf representations of F over GF(p) have the same degree, namely the size of the multiplicative group generated by p, r, and k modulo m.

  We determine the degree of the irreducible fpf representation of the smallest, not metacyclic type-2-group (order 120) over the field GF(7):

          gap> DegreeOfIrredFpfRep2( 7, 30, 11, -1 );
          8

- `DegreeOfIrredFpfRep3(` p`,` m`,` r`)`

  returns the degree of the irreducible fpf representations of the type-III-group F determined by parameters m and r over GF(p). If the parameters are not feasible, then an error is returned. See `FpfRepresentations3` for a presentation of this group.

  All irreducible fpf representations of this group over GF(p) have the same degree.

  We determine the degree of the irreducible fpf representation of SL(2,3) over GF(5):

          gap> DegreeOfIrredFpfRep3( 5, 3, 1 );
          2

- `DegreeOfIrredFpfRep4(` p`,` m`,` r`,` k`)`

  returns the degree of the irreducible fpf representations of the type-IV-group F determined by parameters m, r, and k over GF(p). If the parameters are not feasible, then an error is returned. See `FpfRepresentations4` for a presentation of this group.

  All irreducible fpf representations of F over GF(p) have the same degree.

  We determine the degree of the irreducible fpf representation of the smallest type-4-group, the binary octahedral group of order 48, over GF(5):

          gap> DegreeOfIrredFpfRep4( 5, 3, 1, -1 );
          4

- `FpfRepresentationsCyclic(` p`,` m`)`

  Let a generate a cyclic group of order m. For p and m coprime `FpfRepresentationsCyclic` returns a list of matrices { Ai | i in indexlist } over GF(p) as well as the list indexlist. For all i in indexlist the representation a ® Ai is irreducible and fpf. The Ai with i in indexlist describe all irreducible fpf representations up to equivalence; each irreducible fpf representation is equivalent to one a ® Ai and no two representations a ® Ai, a ® Aj with i ¹ j and i,j in indexlist are equivalent.

  Note, that every faithful irreducible representation of a cyclic group is fpf. The number of nonequivalent faithful irreducible representations over GF(p) is given as f(m)/d, where the degree d is given as the multiplicative order of p modulo m and f(m) denotes the number of residues coprime to m.

  We determine the irreducible matrix representations of the cyclic group of size 8 over GF(5):

          gap> aux := FpfRepresentationsCyclic( 5, 8 );
          [ [ [ [ Z(5)^3, Z(5)^2 ], [ Z(5), Z(5) ] ],
                [ [ Z(5)^2, Z(5) ], [ Z(5)^0, Z(5)^0 ] ] ], [ 1, 7 ] ]
          gap> mats := aux[1];
          [ [ [ Z(5)^3, Z(5)^2 ], [ Z(5), Z(5) ] ],
            [ [ Z(5)^2, Z(5) ], [ Z(5)^0, Z(5)^0 ] ] ]
          gap> indexlist := aux[2];
          [ 1, 7 ]

- `FpfRepresentationsMetacyclic(` p`,` m`,` r`)`

  Let F be a metacyclic group (i.e., F has a cyclic normal subgroup with cyclic factor) admitting an fpf representation. Then F fulfills one of the following two presentations, I or II. Both presentations are determined by integers m and r satisfying certain conditions:

  Type I. Presentation of an fpf metacyclic group F with all Sylow subgroups cyclic. Let m and r satisfy the following conditions:

  (a)
    m and r are coprime.
  (b)
    Let n be the multiplicative order of r modulo m. Then each prime divisor of n divides m.
  (c)
    Let m¢ be maximal such that m¢ divides m and m¢ is coprime to n. Then r = 1 mod (m/m¢).

  Type II. Presentation of an fpf metacyclic group F with generalized quaternion 2-Sylow subgroup. Let m and r satisfy the following conditions:

  (a)
    m and r are coprime.
  (b)
    Let n be the multiplicative order of r modulo m. Then n is 2 times an odd number and each prime divisor of n divides m.
  (c)
    Let s be maximal such that 2s divides m. Then 2s ³ 4 and r = -1 mod 2s.
  (d)
    Let m¢ be maximal such that m¢ divides m/2 and m¢ is coprime to n/2. Then r = 1 mod (m/m¢).

  Then the group F with 2 generators a,b satisfying the relations am = 1, bn = am¢, b-1ab = ar is metacyclic and fpf and has size mn.

  A group satisfying presentation I is of type I in the notation of Zassenhaus, Wolf; presentation II gives a type-II-group.

  Let m, r be as above, and let p coprime to m. Additionally, we require that m does not divide r-1. (Otherwise, F = áa,b | am = 1, bn = am¢, b-1ab = ar ñ would be cyclic.) Then `FpfRepresentationsMetacyclic` returns a list of matrices { (Ai,Bi) | i in indexlist } over GF(p) as well as the list indexlist. The GF(p)-representations determined by a ® Ai and b ® Bi are all irreducible and fpf representations of F = áa,b | am = 1, bn = am¢, b-1ab = ar ñ up to equivalence.

  We determine the irreducible matrix representation of the quaternion group (parameters m = 4, r = -1) over GF(7):

          gap> aux := FpfRepresentationsMetacyclic( 7, 4, -1 );
          [ [ [ [ [ Z(7)^2, Z(7) ], [ Z(7), Z(7)^5 ] ],
                    [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, 0*Z(7) ] ] ] ], [ 1 ] ]
          gap> mats := aux[1];
          [ [ [ [ Z(7)^2, Z(7) ], [ Z(7), Z(7)^5 ] ],
                [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, 0*Z(7) ] ] ] ]

- `FpfRepresentations2(` p`,` m`,` r`,` k`)`

  The presentation of a type-II-group which is not metacyclic is determined by integers m,r,k satisfying the following conditions:

  (a)
    m and r are coprime, m and k are coprime.
  (b)
    Let n be the multiplicative order of r modulo m. Then n is 2 times an odd number and each prime divisor of n divides m.
  (c)
    Let m¢ be maximal such that m¢ divides m and m¢ is coprime to n. Then r = 1 mod (m/m¢).
  (d)
    Let 2s-1 be maximal such that 2s-1 divides m. Define l = -1 mod 2s-1 and l = 1 mod (nm/(2s-1m¢)). Then k = l mod (m/m¢).
  (e)
    The multiplicative order of k modulo m equals 2 and k ¹ r(n/2) mod m.

  Then the group F with generators a,b,q satisfying the relations am = 1, bn = am¢, b-1ab = ar and furthermore q-1a q = ak, q-1b q = bl is fpf of type II and has size 2mn.

  a,b generate a metacyclic group with all Sylow subgroups cyclic (see conditions (a), (b), (c)) of index 2 in F.

  For m, r, k as above and p coprime to m `FpfRepresentations2` returns a list of matrices { (Ai,Bi,Qi) | i in indexlist } over GF(p) as well as the list indexlist. The GF(p)-representations determined by a ® Ai, b ® Bi and q ® Qi are all irreducible, fpf representations of F upto equivalence.

  We determine the irreducible matrix representations of the smallest type-II-group which is not metacyclic (parameters m = 30, r = 11, k = -1, size 120) over the field GF(11) and obtain 2 nonequivalent fpf representations, each of degree 4:

          gap> DegreeOfIrredFpfRep2( 11, 30, 11, -1 );
          4
          gap> aux := FpfRepresentations2( 11, 30, 11, -1 );
          [ [ [ <block matrix of dimensions (2*2)x(2*2)>,
                    <block matrix of dimensions (2*2)x(2*2)>,
                    <block matrix of dimensions (2*2)x(2*2)> ],
                [ <block matrix of dimensions (2*2)x(2*2)>,
                    <block matrix of dimensions (2*2)x(2*2)>,
                    <block matrix of dimensions (2*2)x(2*2)> ] ], [ 1, 13 ] ]

- `FpfRepresentations3(` p`,` m`,` r`)`

  A group F admitting an fpf representation is said to be of type III if F is the semidirect product of the quaternion group and a metacyclic fpf group H of odd size, with the quaternion group normal and H permuting the 3 subgroups of order 4.

  The presentation of a type-III-group is determined by integers m and r, describing the metacyclic group H and its action on the normal quaternion subgroup. The following conditions have to be satisfied for m,r:

  (a)
    3 divides m; m is odd; m and r are coprime.
  (b)
    Let n be the multiplicative order of r modulo m. Then each prime divisor of n divides m.
  (c)
    Let m¢ be maximal such that m¢ divides m and m¢ is coprime to n. Then r = 1 mod (m/m¢).

  Let p,q with relations p4 = 1, q2 = p2, q-1pq = p-1 generate the quaternion group. Let a,b generate a metacyclic group determined by m and r (See `FpfRepresentationsMetacyclic`).

  If 3 divides n, then let a commute with p,q and let b-1pb = q, b-1qb = pq.

  If 3 does not divide n, then let b commute with p,q and let a-1pa = q, a-1qa = pq

  Then the group F with generators p,q,a,b is of type III and has size 8mn.

  For r ¹ 1 mod m, `FpfRepresentations3` returns a list of matrices { (P, Q, Ai,Bi) | i in indexlist } over GF(p) as well as the list indexlist.

  For r = 1 mod m, the group H is cyclic and `FpfRepresentations3` returns { (P, Q, Ai) | i in indexlist } over GF(p) and indexlist.

  The GF(p)-representations determined by p ® P, q ® Q and a ® Ai, b ® Bi are all irreducible, fpf representations of F upto equivalence.

  We determine the irreducible matrix representation of the smallest type-III-group, namely SL(2,3), (parameters m = 3, r = 1, size 24) over the field GF(5):

          gap> aux := FpfRepresentations3( 5, 3, 1 );
          [ [ [ [ [ Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^3 ] ],
                    [ [ 0*Z(5), Z(5)^2 ], [ Z(5)^0, 0*Z(5) ] ],
                    [ [ Z(5)^3, Z(5)^0 ], [ Z(5), Z(5)^0 ] ] ] ], [ 1 ] ]

- `FpfRepresentations4(` p`,` m`,` r`,` k`)`

  A group F = áp,q,a,b,zñ admitting an fpf representation is said to be of type IV, if it has a normal subgroup H = áp,q,a,bñ of type III and index 2.

  The presentation of a type-IV-group is determined by integers m,r,k satisfying the following conditions:

  (a)
    Let s be maximal such that 3s divides m. Then s ³ 1; m is odd; m and r are coprime.
  (b)
    Let n be the multiplicative order of r modulo m. Then 3 does not divide n; each prime divisor of n divides m.
  (c)
    Let m¢ be maximal such that m¢ divides m and m¢ is coprime to n. Then r = 1 mod (m/m¢).
  (d)
    k = -1 mod 3s, k = 1 mod (m/m¢) and k2 = 1 modulo m.

  Let p,q,a,b generate a type-III-group determined by m,r with relations as given in Section `FpfRepresentations3`. Additionally, let z2 = p2, z-1pz = qp, z-1qz = q-1 and z-1a z = ak,z-1b z = b.

  Then the group F with generators p,q,a,b and z is of type IV and has size 16mn.

  For r ¹ 1 mod m, `FpfRepresentations4` returns a list of matrices { (P, Q, Ai,Bi, Zi) | i in indexlist } over GF(p) as well as the list indexlist.

  For r = 1 mod m, the function `FpfRepresentations4` returns { (P, Q, Ai, Zi) | i in indexlist } over GF(p) and indexlist.

  The GF(p)-representations determined by p ® P, q ® Q and a ® Ai, b ® Bi, z ® Zi are all irreducible, fpf representations of F upto equivalence.

  We determine the 2 nonequivalent irreducible matrix representations of the smallest type-IV-group (binary octahedral group, m = 3, r = 1, k = -1, size 48) over the field GF(7):

          gap> aux := FpfRepresentations4( 7, 3, 1, -1 );
          [ [ [ [ [ Z(7)^2, Z(7) ], [ Z(7), Z(7)^5 ] ],
                    [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, 0*Z(7) ] ],
                    [ [ Z(7)^2, 0*Z(7) ], [ Z(7)^0, Z(7)^4 ] ],
                    [ [ Z(7)^5, Z(7) ], [ Z(7), Z(7)^2 ] ] ],
                [ [ [ Z(7)^2, Z(7) ], [ Z(7), Z(7)^5 ] ],
                    [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, 0*Z(7) ] ],
                    [ [ Z(7)^2, 0*Z(7) ], [ Z(7)^0, Z(7)^4 ] ],
                    [ [ Z(7)^2, Z(7)^4 ], [ Z(7)^4, Z(7)^5 ] ] ] ],
            [ [ 1, 1 ], [ -1, 1 ] ] ]

  ## 9.3 Fixed-point-free automorphism groups

- `FpfAutomorphismGroupsCyclic(` ints`,` m`)`

  If `AbelianGroup(`ints`)` admits a cyclic fpf automorphism group of size m, then `FpfAutomorphismGroupsCyclic` determines one representative for each conjugacy class of such fpf automorphism groups. Conjugacy is determined within the whole automorphism group of `AbelianGroup(`ints`)`.

  ints has to be a list of prime power integers and is sorted in the function, according to the order pi £ qj Û p < q or (p = q and j < i).

  `AbelianGroup(`ints`)` admits a cyclic fpf automorphism group of size m iff the multiplicity of each prime power pi in ints is divisible by `DegreeOfIrredFpfRepCyclic( p, m )`.

  A list of generators of the nonconjugate fpf automorphism groups is returned together with the group `AbelianGroup(`ints`)`, on which the automorphisms act. Here ints is sorted with the order above.

  The generators, as, of the 2 nonconjugate cyclic fpf automorphism groups of order 4 on Z25×Z5 are computed as follows:

          gap> aux := FpfAutomorphismGroupsCyclic( [25,5], 4 );
          [ [ [ f1, f3 ] -> [ f1^2*f2, f3^2 ],
                [ f1, f3 ] -> [ f1^2*f2, f3^3 ] ],
            <pc group of size 125 with 2 generators> ]
          gap> as := aux[1];
          [ [ f1, f3 ] -> [ f1^2*f2, f3^2 ], [ f1, f3 ] -> [ f1^2*f2, f3^3 ] ]
          gap> G := aux[2];
          <pc group of size 125 with 2 generators>

- `FpfAutomorphismGroupsMetacyclic(` ints`,` m`,` r`)`

  If `AbelianGroup(`ints`)` admits a metacyclic fpf automorphism group determined by parameters m and r that is not cyclic (see `FpfRepresentationsMetacyclic` for a presentation), then `FpfAutomorphismGroupsMetacyclic` determines one representative for each conjugacy class of such fpf automorphism groups. Conjugacy is determined within the whole automorphism group of `AbelianGroup(`ints`)`.

  ints has to be a list of prime power integers and is sorted in the function, according to the order pi £ qj Û p < q or (p = q and i ³ j).

  Moreover, the multiplicity of each prime power pi in ints has to be divisible by `DegreeOfIrredFpfRepMetacyclic( p, m, r )`, which is a multiple of the multiplicative order of r modulo m.

  A list of pairs of generators (a,b satisfying b-1ab = ar, am = 1 and bn = am¢) of the nonconjugate fpf automorphism groups is returned together with the group `AbelianGroup(`ints`)`, on which the automorphisms act. Here ints is sorted with the order above.

  For G = (Z3)2×(Z5)2 the quaternion fpf automorphism group of size 8 (parameters m = 4, r = -1) is computed as follows:

          gap> aux := FpfAutomorphismGroupsMetacyclic( [3,3,5,5], 4, -1 );
          [ [ [ [ f1, f2, f3, f4 ] -> [ f1^2, f2^3, f3*f4, f3*f4^2 ],
                    [ f1, f2, f3, f4 ] -> [ f2^4, f1, f4^2, f3 ] ] ],
            <pc group of size 225 with 4 generators> ]
          gap> fs := aux[1];
          [ [ [ f1, f2, f3, f4 ] -> [ f1^2, f2^3, f3*f4, f3*f4^2 ],
                [ f1, f2, f3, f4 ] -> [ f2^4, f1, f4^2, f3 ] ] ]
          gap> phi := Group( fs[1] );
          <group with 2 generators>
          gap> G := aux[2];
          <pc group of size 225 with 4 generators>

  On G = (Z7)2×(Z17)2 there are 2 nonconjugate fpf automorphism groups isomorphic to the generalized quaternion group of size 16 (parameters m = 8, r = -1):

          gap> aux := FpfAutomorphismGroupsMetacyclic( [7,7,17,17], 8, -1 );;
          gap> fs := aux[1];
          [ [ [ f1, f2, f3, f4 ] -> [ f1^9, f2^2, f3^4*f4^2, f3*f4^6 ],
                [ f1, f2, f3, f4 ] -> [ f2^16, f1, f3^4*f4^5, f3^5*f4^3 ] ],
            [ [ f1, f2, f3, f4 ] -> [ f1^9, f2^2, f3^3*f4^5, f3^6*f4 ],
                [ f1, f2, f3, f4 ] -> [ f2^16, f1, f3^3*f4^4, f3*f4^4 ] ] ]
          gap> phis := List( fs, Group );
          [ <group with 2 generators>, <group with 2 generators> ]
          gap> G := aux[2];
          <pc group of size 14161 with 4 generators>

- `FpfAutomorphismGroups2(` ints`,` m`,` r`,` k`)`

  If `AbelianGroup(`ints`)` admits an fpf automorphism group of type II, determined by parameters m, r, k that is not metacyclic (see `FpfRepresentations2` for a presentation), then `FpfAutomorphismGroups2` determines one representative for each conjugacy class of such fpf automorphism groups. Conjugacy is determined within the whole automorphism group of `AbelianGroup(`ints`)`.

  ints has to be a list of prime power integers and is sorted in the function, according to the order pi £ qj Û p < q or (p = q and i ³ j).

  Note, that the degree of an irreducible fpf representation of a type-II-group which is not metacyclic is divisible by 4 and that the multiplicity of each prime power pi in ints has to be divisible by `DegreeOfIrredFpfRep2( p, m, r, k )`.

  A list of triples of generators (a,b,z satisfying b-1ab = ar, am = 1 and z-1az = ak) of the nonconjugate fpf automorphism groups is returned together with the group `AbelianGroup(`ints`)`, on which the automorphisms act. Here ints is sorted with the order above.

  Upto conjugacy there is only one fpf automorphism group of type II with parameters m = 30, r = 11, k = -1, size 120 on the elementary abelian group of size 114:

          gap> aux := FpfAutomorphismGroups2( [11,11,11,11], 30, 11, -1 );
          [ [ [ [ f1, f2, f3, f4 ] -> [ f1^5*f2^4, f1^3*f2^10, f3^2*f4^8,
                        f3^6*f4 ],
                    [ f1, f2, f3, f4 ] -> [ f1^3*f2^10, f1^10*f2^8, f3^8*f4,
                        f3*f4^3 ],
                    [ f1, f2, f3, f4 ] -> [ f3^10, f4^10, f1, f2 ] ] ],
            <pc group of size 14641 with 4 generators> ]
          gap> phi := Group( aux[1][1] );
          <group with 3 generators>
          gap> G := aux[2];
          <pc group of size 14641 with 4 generators>

- `FpfAutomorphismGroups3(` ints`,` m`,` r`)`

  If `AbelianGroup(`ints`)` admits an fpf automorphism group of type III determined by parameters m and r (see `FpfRepresentations3` for a presentation), then `FpfAutomorphismGroups3` determines one representative for each conjugacy class of such fpf automorphism groups. Conjugacy is determined within the whole automorphism group of `AbelianGroup(`ints`)`.

  ints has to be a list of prime power integers and is sorted in the function, according to the order pi £ qj Û p < q or (p = q and i ³ j).

  Moreover, the multiplicity of each prime power pi in ints has to be divisible by `DegreeOfIrredFpfRep3( p, m, r )`, which is a multiple of 2n where n is the multiplicative order of r modulo m.

  A list of tuples of generators, [p,q,a,b], (p,q generating the quaternion group, a,b satisfying b-1ab = ar, am = 1 and bn = am¢) of the nonconjugate fpf automorphism groups is returned together with the group `AbelianGroup(`ints`)`, on which the automorphisms act. Here ints is sorted with the order above.

  For G = (Z5)2 the fpf automorphism type-III-group isomorphic to SL(2,3) is computed as follows (parameters m = 3, r = 1):

          gap> aux := FpfAutomorphismGroups3( [5,5], 3, 1 );
          [ [ [ [ f1, f2 ] -> [ f1^2, f2^3 ], [ f1, f2 ] -> [ f2^4, f1 ],
                    [ f1, f2 ] -> [ f1^3*f2, f1^2*f2 ] ] ],
            <pc group of size 25 with 2 generators> ]
          gap> phi := Group( aux[1][1] );
          <group with 3 generators>
          gap> G := aux[2];
          <pc group of size 25 with 2 generators>

- `FpfAutomorphismGroups4(` ints`,` m`,` r`,` k`)`

  If `AbelianGroup(`ints`)` admits an fpf automorphism group of type IV determined by parameters m, r, k (see `FpfRepresentations4` for a presentation), then `FpfAutomorphismGroups4` determines one representative for each conjugacy class of such fpf automorphism groups. Conjugacy is determined within the whole automorphism group of `AbelianGroup(`ints`)`.

  ints has to be a list of prime power integers and is sorted in the function, according to the order pi £ qj Û p < q or (p = q and i ³ j).

  Moreover, the multiplicity of each prime power pi in ints has to be divisible by `DegreeOfIrredFpfRep4( p, m, r )`, which is a multiple of 2n where n is the multiplicative order of r modulo m.

  A list of tuples of generators, [p,q,a,b,z], of the nonconjugate fpf automorphism groups is returned together with the group `AbelianGroup(`ints`)`, on which the automorphisms act. Here ints is sorted with the order above. If r = 1 mod m, then a list of tuples, [p,q,a,z], is returned instead.

  For G = (Z7)2 the fpf automorphism type-IV-group isomorphic the binary octahedral group of size 48 (parameters m = 3, r = 1, k = -1) is computed as follows:

          gap> aux := FpfAutomorphismGroups4( [7,7], 3, 1, -1 );
          [ [ [ [ f1, f2 ] -> [ f1^2*f2^3, f1^3*f2^5 ],
                    [ f1, f2 ] -> [ f2^6, f1 ], [ f1, f2 ] -> [ f1^2, f1*f2^4 ],
                    [ f1, f2 ] -> [ f1^5*f2^3, f1^3*f2^2 ] ] ],
            <pc group of size 49 with 2 generators> ]
          gap> phi := Group( aux[1][1] );
          <group with 4 generators>
          gap> G := aux[2];
          <pc group of size 49 with 2 generators>

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