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

Returns the dual group of a given finite abelian group.

<p><p>

<h3>
Syntax:
</h3>
<pre>d := AbelianDualGroup(g);</pre>

<p>

<table>
<tr>
<td>groups</td>
<td><pre>  g, d  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
Returns the dual group d of a group g, i.e. the group
consisting of the isomorphisms from the group g to C.
The isomorphisms are presented by rational numbers q
corresponding to e^{(i*2*\pi*q)}.


<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Compute the dual group and apply one homomorphism: 
<pre>

kash> g := AbelianGroupCreate([[0,1,2],[5,6,0],[0,4,5]]);
Group with relations:
[0 1 2]
[5 6 0]
[0 4 5]
kash> d := AbelianDualGroup(g);
Group with relations:
[15]
kash> eltd := AbelianGroupEltCreate(d, [1]);
[1]
kash> hom := AbelianGroupDiscreteExp(eltd);
HomMatrix =
[ -6   6]
[ 10 -10]
[ -5   5] 
from Group with relations:
[0 1 2]
[5 6 0]
[0 4 5] 
to Group with relations:
[3 0]
[0 5]

kash> eltg := AbelianGroupEltCreate(g, [1,0,0]);
[1 0 0]
kash> AbelianGroupDiscreteExp(hom*eltg);
1/5


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
 It is allowed to apply an elelment of the group to
an element of the corresponding dual group: 
<pre>

kash> eltd*eltg;
> 1/5

</pre>

<p>


<hr>

<p>
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