[AutPGrp] 3 The underlying function

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# 3 The underlying function

Underlying the method installation for `AutomorphismGroup` is the function `AutomorphismGroupPGroup`. This function is intended for expert users who wish to influence the steps of the algorithm. Note also that `AutomorphismGroup` will always choose default values.

- `AutomorphismGroupPGroup(` G`[,`flag`] ) F`

  The input is a finite p-group as above and an optional flag which can be true or false. Here the filters for G need not be set, but they should be true for G. The possible values for flag are considered later in Chapter Influencing the algorithm. If flag is not supplied, the algorithm proceeds similarly to the method installed for `AutomorphismGroup`, but it produces slightly more detailed output. The output of the function is a record which contains the following fields:

  `glAutos`
    a set of automorphisms which together with `agAutos` generate the automorphism group;

  `glOrder`
    an integer whose product with the `agOrders` gives the size of the automorphism group;

  `agAutos`
    a polycyclic generating sequence for a soluble normal subgroup of the automorphism group;

  `agOrder`
    the relative orders corresponding to `agAutos`;

  `one`
    the identity element of the automorphism group;

  `group`
    the underlying group G;

  `size`
    the size of the automorphism group.

  We do not return an automorphism group in the standard form because we wish to distinguish between `agAutos` and `glAutos`; the latter act non-trivially on the Frattini quotient of G. This hybrid-group description of the automorphism group permits more efficient computations with it. The following function converts the output of `AutomorphismGroupPGroup` to the output of `AutomorphismGroup`.

- `ConvertHybridAutGroup(` A`) F`

      gap> LoadPackage("autpgrp", false);
      true

      gap> H := SmallGroup (729, 34);
      <pc group of size 729 with 6 generators>

      gap> A := AutomorphismGroupPGroup(H);
      rec( glAutos := [  ],
           glOrder := 1,
           agAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6 ])
                          -> [ f1^2, f2, f3^2*f4, f4, f5^2*f6, f6 ],
                        Pcgs([ f1, f2, f3, f4, f5, f6 ])
                          -> [ f2^2, f1, f3*f5^2, f5^2, f4*f6^2, f6^2 ],
                        Pcgs([ f1, f2, f3, f4, f5, f6 ])
                          -> [ f1^2, f2^2, f3*f4^2*f5^2*f6, f4^2*f6, f5^2*f6, f6 ],
                        Pcgs([ f1, f2, f3, f4, f5, f6 ])
                          -> [ f1*f3, f2, f3*f5^2, f4*f6^2, f5, f6 ],
                        Pcgs([ f1, f2, f3, f4, f5, f6 ])
                          -> [ f1, f2*f3, f3*f4, f4, f5*f6, f6 ],
                        Pcgs([ f1, f2, f3, f4, f5, f6 ])
                          -> [ f1*f4, f2, f3*f6^2, f4, f5, f6 ],
                        Pcgs([ f1, f2, f3, f4, f5, f6 ])
                          -> [ f1, f2*f4, f3, f4, f5, f6 ],
                        Pcgs([ f1, f2, f3, f4, f5, f6 ])
                          -> [ f1*f5, f2, f3, f4, f5, f6 ],
                        Pcgs([ f1, f2, f3, f4, f5, f6 ])
                          -> [ f1, f2*f5, f3*f6, f4, f5, f6 ],
                        Pcgs([ f1, f2, f3, f4, f5, f6 ])
                          -> [ f1*f6, f2, f3, f4, f5, f6 ],
                        Pcgs([ f1, f2, f3, f4, f5, f6 ])
                          -> [ f1, f2*f6, f3, f4, f5, f6 ] ],
           agOrder := [ 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3 ],
           one     := IdentityMapping( <pc group of size 729 with 6 generators> ),
           group   := <pc group of size 729 with 6 generators>,
           size    := 52488 )

      gap> ConvertHybridAutGroup( A );
      <group of size 52488 with 11 generators>

  Let A be the automorphism group of a p-group G as computed by `AutomorphismGroupPGroup`. Then the following function can compute a pc group isomorphic to the solvable part of A stored in the record component A.agGroup. This solvable part forms a subgroup of the automorphism group which contains at least the automorphisms centralizing the Frattini factor of G. The pc group facilitates various further computations with A.

- `PcGroupAutPGroup(` A`) F`

  computes a pc presentation for the solvable part of the automorphism group A defined by A.agGroup. A is the output of the function `AutomorphismGroupPGroup`.

      gap> H := SmallGroup (729, 34);;
      gap> A := AutomorphismGroupPGroup(H);;
      gap> B := PcGroupAutPGroup( A );
      <pc group of size 52488 with 11 generators>
      gap> I := InnerAutGroupPGroup( B );
      Group([ f5, f4^2*f8, f6^2*f9^2, f11^2, f10^2, <identity> of ... ])

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