GAP (AutPGrp) - Chapter 2: 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

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

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

2 The automorphism group method

2.1 The automorphism group method

2.1-1 AutomorphismGroup

2.1-2 InfoAutGrp