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<html><head><title>[AutPGrp] 3 The underlying function</title></head>
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<h1>3 The underlying function</h1><p>
<p>
Underlying the method installation for <code>AutomorphismGroup</code>
is the function <code>AutomorphismGroupPGroup</code>. This function is
intended for expert users who wish to influence the steps of 
the algorithm.  Note also that <code>AutomorphismGroup</code> will always
choose default values.
<p>
<a name = ""></a>
<li><code>AutomorphismGroupPGroup( </code><var>G</var><code> [,</code><var>flag</var><code>] ) F</code>
<p>
The input is a finite <var>p</var>-group as above and an optional <var>flag</var> 
which can be true or false. Here the filters for <var>G</var> need not be 
set, but they should be true for <var>G</var>. The possible values for <var>flag</var>
are considered later in Chapter <a href="CHAP004.htm">Influencing the algorithm</a>. If 
<var>flag</var> is not supplied, the algorithm proceeds similarly to the 
method installed for <code>AutomorphismGroup</code>, but it produces slightly 
more detailed output.  The output of the function is a record 
which contains the following fields:
<p>
<p>
<dl compact>
<dt><code>glAutos</code> <dd> a set of automorphisms which together with <code>agAutos</code>
            generate the automorphism group;
<p>
<dt><code>glOrder</code> <dd> an integer whose product with the <code>agOrders</code> gives
            the size of the automorphism group;
<p>
<dt><code>agAutos</code> <dd> a polycyclic generating sequence for a soluble normal
            subgroup of the automorphism group;
<p>
<dt><code>agOrder</code> <dd> the relative orders corresponding to <code>agAutos</code>;
<p>
<dt><code>one</code>     <dd> the identity element of the automorphism group;
<p>
<dt><code>group</code>   <dd> the underlying group <var>G</var>;
<p>
<dt><code>size</code>    <dd> the size of the automorphism group.
</dl>
<p>
We do not return an automorphism group in the standard form 
because we wish to distinguish between <code>agAutos</code> and <code>glAutos</code>; 
the latter act non-trivially on the Frattini quotient of <var>G</var>. This
hybrid-group description of the automorphism group permits more 
efficient computations with it. The following function converts
the output of <code>AutomorphismGroupPGroup</code> to the output of 
<code>AutomorphismGroup</code>.
<p>
<a name = ""></a>
<li><code>ConvertHybridAutGroup( </code><var>A</var><code> ) F</code>
<p>
<pre>
gap&gt; RequirePackage("autpgrp");
#I ------------ The AutPGrp package --------------
#I -- Computing automorphism groups of p-groups -- 
true

gap&gt; H := SmallGroup (729, 34);
&lt;pc group of size 729 with 6 generators&gt;

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

gap&gt; ConvertHybridAutGroup( A );
&lt;group of size 52488 with 11 generators&gt;
</pre>
<p>
<p>
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<P>
<address>AutPGrp manual<br>August 2003
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