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<html><head><title>[AutPGrp] 2 The automorphism group method</title></head>
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<h1>2 The automorphism group method</h1><p>
<p>
The <font face="Gill Sans,Helvetica,Arial">AutPGrp</font> package installs a method for <code>AutomorphismGroup</code> for a 
finite <var>p</var>-group (see also Section&nbsp;<a href="../../../doc/htm/ref/CHAP038.htm#SECT007">Groups of Automorphisms</a> 
in the <font face="Gill Sans,Helvetica,Arial">GAP</font> Reference Manual).
<p>
<a name = ""></a>
<li><code>AutomorphismGroup( </code><var>G</var><code> ) M</code>
<p>
The input is a finite <var>p</var>-group <var>G</var>. If the filters <code>IsPGroup</code>, 
<code>IsFinite</code> and <code>CanEasilyComputePcgs</code> are set and true for <var>G</var>, 
the method selection of <font face="Gill Sans,Helvetica,Arial">GAP</font>&nbsp;4 invokes this algorithm. 
<p>
The output of the method is an automorphism group, whose generators 
are given in <code>GroupHomomorphismByImages</code> format in terms of their action 
on the underlying group <var>G</var>. 
<p>
<a name = "I0"></a>

<a name = ""></a>
<li><code>InfoAutGrp V</code>
<p>
This is a <font face="Gill Sans,Helvetica,Arial">GAP</font> InfoClass (these are described in Chapter&nbsp;<a href="../../../doc/htm/ref/CHAP007.htm#SECT004">Info Functions</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font> Reference Manual). By assigning an <var>info-level</var>
in the range 1 to 4 via
<p>
<code>SetInfoLevel(InfoAutGrp, </code><var>info-level</var><code>)</code>
<p>
varying levels of information on the progress of 
the computation, will be obtained. 
<p>
<pre>
gap&gt; RequirePackage("autpgrp");
#I ------------ The AutPGrp package --------------
#I -- Computing automorphism groups of p-groups -- 
true

gap&gt; G := SmallGroup( 32, 15 );
&lt;pc group of size 32 with 5 generators&gt;

gap&gt; SetInfoLevel( InfoAutGrp, 1 );

gap&gt; 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
&lt;group of size 64 with 6 generators&gt;
</pre>
<p>
The algorithm proceeds by induction down the lower <var>p</var>-central
series of <var>G</var> 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.
<p>
<pre>
gap&gt; 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&gt; IsPGroup(G);
true
gap&gt; CanEasilyComputePcgs(G);
true
gap&gt; IsFinite(G);
true
gap&gt; 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
&lt;group of size 128 with 7 generators&gt;
gap&gt; A := last;;
gap&gt; 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) ]) -&gt; 
[ ( 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&gt; Order(A.1);
16
</pre>
<p>
<p>
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<address>AutPGrp manual<br>August 2003
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