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

Tests whether a function is a group automorphism.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := AbelianGroupIsAut(hom [, check]);</pre>

<p>

<table>
<tr>
<td>homomorphism</td>
<td><pre>  hom  </pre></td>
<td></td>
</tr>
<tr>
<td>boolean</td>
<td><pre>  check  </pre></td>
<td></td>
</tr>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
Tests whether the function <tt>hom</tt> is an automorphism.
<tt> AbelianGroupIsAut</tt> returns the list <tt>[true, hom^{-1</tt>]}
iff <tt>hom</tt> is an automorphism, and the list
<tt>[false]</tt> otherwise.
If the second (optional) parameter <tt>check</tt> is <tt>false</tt>
then there is no check, whether <tt>hom</tt> is an endomorphism.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>

<pre>

kash> g := AbelianGroupCreate(Mat(Z, [[2,2], [1,3]]));;
kash> mat := Mat(Z, [[0,2],[0,2]]);;
kash> hom := AbelianGroupHomCreate(g, g, mat, true);;
kash> AbelianGroupIsAut(hom);
[ false ]


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
 Further information: 
<pre>

kash> AbelianGroupPrintLevel := 1;;
kash> AbelianGroupIsAut(hom);
homomorphism is not surjective
[ false ]
kash> AbelianGroupPrintLevel := 0;;


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
 Compute the inverse homomorphism: 
<pre>

kash> mat := Mat(Z, [[-1,2],[3,2]]);;
kash> hom := AbelianGroupHomCreate(g, g, mat, true);;
kash> hominv := hom^-1;
> HomMatrix =
[  0 1/5]
[  0 1/5] 
from Group with relations:
[2 2]
[1 3] 
to Group with relations:
[2 2]
[1 3]


</pre>

<p>


<hr>

<p>
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