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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%A method.tex AutPGrp documentation Bettina Eick
%A Eamonn O'Brien
%%
%H @(#)$Id: method.tex,v 1.2 2002/11/27 07:27:27 gap Exp $
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Chapter{The automorphism group method}
The {\AutPGrp} package installs a method for `AutomorphismGroup' for a
finite $p$-group (see also Section~"ref:Groups of Automorphisms"
in the {\GAP} Reference Manual).
\> AutomorphismGroup( <G> ) M
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>.
\indextt{SetInfoLevel}
\>`InfoAutGrp' V
This is a {\GAP} InfoClass (these are described in Chapter~"ref:Info
Functions" in the {\GAP} Reference Manual). By assigning an <info-level>
in the range 1 to 4 via
\){\kernttindent}SetInfoLevel(InfoAutGrp, <info-level>)
varying levels of information on the progress of
the computation, will be obtained.
\beginexample
gap> RequirePackage("autpgrp");
#I ------------ The AutPGrp package --------------
#I -- Computing automorphism groups of p-groups --
true
gap> G := 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>
\endexample
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.
\beginexample
gap> G := DihedralGroup( IsPermGroup, 2^5 );
Group([ ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16),
( 2,16)( 3,15)( 4,14)( 5,13)( 6,12)( 7,11)( 8,10) ])
gap> IsPGroup(G);
true
gap> CanEasilyComputePcgs(G);
true
gap> IsFinite(G);
true
gap> 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 := last;;
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
\endexample
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