GAP (AutPGrp) - Chapter 2: The automorphism group method

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2 The automorphism group method
2.1 The automorphism group method
2.1-1 AutomorphismGroup
2.1-2 InfoAutGrp

### 2 The automorphism group method

#### 2.1 The automorphism group method

The AutPGrp package installs a method for `AutomorphismGroup` for a finite \(p\)-group (see also Section Reference: Groups of Automorphisms in the GAP Reference Manual).

##### 2.1-1 AutomorphismGroup

`‣ AutomorphismGroup`( G ) | ( operation )

Returns: The automorphism group of G.

The input is a finite \(p\)-group G. If the filters `IsPGroup`, `IsFinite` and `CanEasilyComputePcgs` are set and true for G, the method selection of GAP 4 invokes this algorithm.

The output of the method is an automorphism group, whose generators are given in `GroupHomomorphismByImages` format in terms of their action on the underlying group G.

##### 2.1-2 InfoAutGrp

`‣ InfoAutGrp` | ( info class )

This is a GAP `InfoClass` (Reference: InfoClass for a GAP package). By assigning an `level` in the range 1 to 4 via

    SetInfoLevel(InfoAutGrp, level);

varying levels of information on the progress of the computation, will be obtained.

    gap> LoadPackage("autpgrp", false);
    true
    gap> G := PcGroupCode(619031068735, 32);  # SmallGroup( 32, 15 );
    <pc group of size 32 with 5 generators>
    gap> SetInfoLevel( InfoAutGrp, 1 );
    gap> AutomorphismGroup(G);
    #I  step 1: 2^2 -- init automorphisms
    #I  step 2: 2^2 -- aut grp has size 2
    #I  step 3: 2^1 -- aut grp has size 32
    #I  final step: convert
    <group of size 64 with 6 generators>

The algorithm proceeds by induction down the lower \(p\)-central series of `G` and the information corresponds to the steps of this induction. In the following example we observe that the method also accepts permutation groups as input, provided they satisfy the required filters.

    gap> G := DihedralGroup( IsPermGroup, 2^5 );;
    gap> IsPGroup(G);
    true
    gap> CanEasilyComputePcgs(G);
    true
    gap> IsFinite(G);
    true
    gap> A := AutomorphismGroup(G);
    #I  step 1: 2^2 -- init automorphisms
    #I  step 2: 2^1 -- aut grp has size 2
    #I  step 3: 2^1 -- aut grp has size 8
    #I  step 4: 2^1 -- aut grp has size 32
    #I  final step: convert
    <group of size 128 with 7 generators>
    gap> A.1;
    Pcgs([ ( 2,16)( 3,15)( 4,14)( 5,13)( 6,12)( 7,11)( 8,10),
      ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16),
      ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16),
      ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16),
      ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ]) ->
    [ ( 1, 2)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10),
      ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16),
      ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16),
      ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16),
      ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ]
    gap> Order(A.1);
    16

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