<html><head><title>[SONATA-tutorial] 4 Nearrings of transformations on groups</title></head>
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<h1>4 Nearrings of transformations on groups</h1><p>
<p>
We are going to study transformations on the alternating group on four elements
<var>A<sub>4</sub></var>.
<p>
<strong>The problem</strong>: Let <var>T</var> be the nearring of mappings from <var>A<sub>4</sub></var> to <var>A<sub>4</sub></var>
generated by the single mapping <var>t</var> which maps (2,3,4) to (2,4,3),
(2,4,3) to (1,2)(3,4), (1,2)(3,4) to (1,2,3), (1,2,3) back to (2,3,4)
and all other elements of <var>A<sub>4</sub></var> to the neutral element (). Then,
how many mappings are there in <var>T</var> that have (1,2,3) as a fixed point?
If there are only a few we would be interested in a list of all of
these.
<p>
<strong>The solution</strong>: <br>
The first thing to do is create the nearring <var>T</var>. So we start with
the group <var>A<sub>4</sub></var>, which can easily be constructed with the command
<pre>
gap> A4 := AlternatingGroup( 4 );
Alt( [ 1 .. 4 ] )
</pre>
The result is an object which represents the group <var>A<sub>4</sub></var>. If we want
to see its elements we have to ask <font face="Gill Sans,Helvetica,Arial">GAP</font> to make a list of elements
out of the group.
<pre>
gap> AsSortedList( A4 );
[ (), (2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2),
(1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3) ]
</pre>
Now we create the mapping <var>t</var>. We use the function
<code>MappingByPositionList</code> to enter it.
<pre>
t := EndoMappingByPositionList( A4, [1,3,4,5,2,1,1,1,1,1,1,1] );
<mapping: AlternatingGroup( [ 1 .. 4 ] ) -> AlternatingGroup(
[ 1 .. 4 ] ) >
</pre>
For <code>Mappings</code> the usual operations <code>+</code> and
<code>*</code> can be used to add and multiply them.
<pre>
gap> t+t;
<mapping: AlternatingGroup( [ 1 .. 4 ] ) -> AlternatingGroup(
[ 1 .. 4 ] ) >
gap> last * t;
<mapping: AlternatingGroup( [ 1 .. 4 ] ) -> AlternatingGroup(
[ 1 .. 4 ] ) >
</pre>
(Recall that <code>last</code> stands for the result of the last computation, in
this case this is <code>t + t</code>).
Now we can construct the nearring. We use the function
<code>TransformationNearRingByGenerators</code> which asks for the group (<var>A<sub>4</sub></var>)
and a list of generating elements (the list with <var>t</var> as the only entry)
as arguments.
<pre>
gap> T := TransformationNearRingByGenerators( A4, [ t ] );;
</pre>
Nearrings, allthough generated by a single element can become rather
big. Before we print out all elements we ask for the size of <var>T</var>.
<pre>
gap> Size( T );
20736
</pre>
It seems reasonable not to print all elements. <strong>Note</strong> that they are
not even computed, yet. All we wanted to know was the size of <var>T</var> and
this can be computed without generating all elements. But, yes, we
could generate them with <code>AsList</code> or <code>AsSortedList</code>. At last we want
to find out how many of these 20736 <code>GroupTransformations</code> have (1,2,3)
as a fixed point. We filter them out, but we use a second semicolon at
the end to suppress printing, because there might be a lot of them.
Then we ask for the length of the resulting list <var>F</var> of mappings.
<pre>
gap> F := Filtered( T, tfm -> Image( tfm, (1,2,3) ) = (1,2,3) );;
gap> Length( F );
1728
</pre>
It seems not to be worth printing the whole list. But we could for
example choose a random transformation from this list <var>F</var> for testing
purposes.
<pre>
gap> Random( F );;
</pre>
There are of course other properties of the nearring <var>T</var>
which might be interesting. It is clear that a nearring which is
generated by a single element is not necessarily abelian. <var>T</var> is a
counterexample. As for finding counterexamples, SONATA can be used
as a research tool.
<pre>
gap> IsCommutative( T );
false
</pre>
Finally, we try to disprove the conjecture that every transformation
nearring on an abelian group that is generated by a single element
must be commutative.
<pre>
gap> g := CyclicGroup(2);;
gap> m := MapNearRing(g);;
gap> Filtered( m, n -> not( IsCommutative(
> TransformationNearRingByGenerators( g, [n] ) ) ) );
gap> [ <mapping: Group( [ f1 ] ) -> Group( [ f1 ] ) >,
<mapping: Group( [ f1 ] ) -> Group( [ f1 ] ) > ]
gap> GraphOfMapping(last[1]);
[ [ <identity> of ..., f1 ], [ f1, <identity> of ... ] ]
</pre>
<p>
<p>
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<P>
<address>SONATA-tutorial manual<br>November 2008
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