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

Returns the canonical quotient group of a homomorphism.

<p><p>

<h3>
Syntax:
</h3>
<pre>q := AbelianGroupCanonicalQuotient(hom);</pre>

<p>

<table>
<tr>
<td>group</td>
<td><pre>  q  </pre></td>
<td></td>
</tr>
<tr>
<td>homomorphism</td>
<td><pre>  hom  </pre></td>
<td></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#AbelianGroupHomImage">AbelianGroupHomImage</a>, <a href="kashref.html#AbelianGroupHomKernel">AbelianGroupHomKernel</a>, <a href="kashref.html#AbelianQuotientGroup">AbelianQuotientGroup</a>

<p>

<h3>
Description:
</h3>
Let <tt>hom</tt> be a homomorphism from a group g to a group h.
Then <tt> AbelianGroupCanonicalQuotient</tt> returns the quotient group
q of g and the kernel of <tt>hom</tt>.
All homomorphisms and isomorphisms
are installed. This is useful if you frequently consider elements of
g, h and the kernel and the image of <tt>hom</tt>.


<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Compute the canonical quotient of a homomorphism: 
<pre>

kash> g := AbelianGroupCreate([[1,2,3],[2,4,0], [2,0,0]]);;
kash> h := AbelianGroupCreate([[1,0,0],[0,4,0], [1,1,1]]);;
kash> mat := Mat(Z, [[0,0,0],[6,0,-6],[4,0,4]]);;
kash> hom := AbelianGroupHomCreate(g, h, mat, true);
HomMatrix =
[ 0  0  0]
[ 6  0 -6]
[ 4  0  4] 
from Group with relations:
[1 2 3]
[2 4 0]
[2 0 0] 
to Group with relations:
[1 0 0]
[0 4 0]
[1 1 1]

kash> q := AbelianGroupCanonicalQuotient(hom);
Group with relations:
[1 0 0]
[0 2 0]
[0 0 1]
[1 2 3]
[2 4 0]
[2 0 0]
kash> AbelianGroupSmithCreate(q);
> Group with relations:
[2]

</pre>

<p>


<hr>

<p>
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