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<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>

We are going to study transformations on the alternating group on four elements
A4.

The problem: Let <var>T</var> be the nearring of mappings from A4 to A4
 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 A4 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.

The solution: 
 The first thing to do is create the nearring <var>T</var>. So we start with
 the group A4, 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 A4. If we want
 to see its elements we have to ask GAP 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 (A4)
 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. Note 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>


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