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<p>&nbsp;<p>
<hr noshade size="5">
<h2><a name="AbelianRayClassGroupAutoCreate"></a>
AbelianRayClassGroupAutoCreate
</h2>

Given a field automorphism, this function generates the
corresponding automorphism of abelian groups.

<p><p>

<h3>
Syntax:
</h3>
<pre>aut := AbelianRayClassGroupAutoCreate(G, sigma);</pre>

<p>

<table>
<tr>
<td>AbelianGroupHom</td>
<td><pre>  aut  </pre></td>
<td></td>
</tr>
<tr>
<td>AbelianGroup</td>
<td><pre>  G  </pre></td>
<td>must be RayClassGroupToAbelianGroup</td>
</tr>
<tr>
<td>KASH function</td>
<td><pre>  sigma  </pre></td>
<td>acting on ideals</td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
<i>no detailed description available yet</i>

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
We demonstrate this in Q(\sqrt{10}):
<pre>

kash> o := OrderMaximal(Z, 2, 10);
Generating polynomial: x^2 - 10
Discriminant: 40 

kash> G := RayClassGroupToAbelianGroup(27*o);
RayClassGroupToAbelianGroup(<27>, [  ])
Group with relations:
[18]
kash> OrderAutomorphisms(o);
[ [0, 1], [0, -1] ]
kash> aut := AbelianRayClassGroupAutoCreate(G, x->IdealAutomorphism(x, 2));
HomMatrix =
[1] 
from RayClassGroupToAbelianGroup(<27>, [  ])
Group with relations:
[18] 
to RayClassGroupToAbelianGroup(<27>, [  ])
Group with relations:
[18]


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
This should be equivalent to (but is faster for larger groups):
<pre>

kash> l := AbelianGroupGenerators(G);
[ [1] ]
kash> Apply(l, AbelianGroupDiscreteExp);
kash> Apply(l, x->IdealAutomorphism(x, 2));
kash> Apply(l, x->AbelianGroupDiscreteLog(G, x));
kash> Apply(l, x->x.expvec);
kash> IsMat(l);
true
kash> aut2 := AbelianGroupHomCreate(G, G, l);
> HomMatrix =
[1] 
from RayClassGroupToAbelianGroup(<27>, [  ])
Group with relations:
[18] 
to RayClassGroupToAbelianGroup(<27>, [  ])
Group with relations:
[18]


</pre>

<p>


<hr>

<p>
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