<html><head><title>[SONATA-tutorial] 7 Planar nearrings</title></head>
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<h1>7 Planar nearrings</h1><p>
<p>
We recall the definition of planar nearrings and basic results
(see citeClay:Nearrings).
Let <var>(N,+,cdot)</var> be a left nearring. For <var>a,b inN</var> we define <var>a equivb</var>
iff <var>acdotn = bcdotn</var> for all <var>ninN</var>. If <var>a equivb</var>, then <var>a</var> and <var>b</var>
are called <strong>equivalent multipliers</strong>.
A nearring <var>N</var> is called <strong>planar</strong> if <var>| N/<sub>equiv</sub> | ge3</var> and if
for any two non-equivalent multipliers <var>a</var> and <var>b</var> in <var>N</var>, for any <var>cinN</var>,
the equation <var>acdotx = bcdotx + c</var> has a unique solution.
<p>
A <strong>Ferrero pair</strong> is a pair of finite groups <var>(N,Phi)</var> such that <var>Phi</var>
is a fixed-point-free automorphism group of <var>(N,+)</var>.
<p>
Starting with a Ferrero pair <var>(N,Phi)</var> we can construct a planar nearring
in the following way:
Select representatives, say <var>e<sub>1</sub>,...,e<sub>t</sub></var>, for some or all of the
non-trivial orbits of <var>N</var> under <var>Phi</var>.
Let <var>C = Phi(e<sub>1</sub>)cup...cupPhi(e<sub>t</sub>)</var>.
For each <var>xinN</var> we define <var>acdotx = 0</var> for <var>ainNsetminusC</var>, and
<var>acdotx=phi<sub>a</sub>(x)</var> for <var>ainPhi(e<sub>i</sub>)subsetC</var> and <var>phi<sub>a</sub>(e<sub>i</sub>)=a</var>.
Then <var>(N,+,cdot)</var> is a (left) planar nearring with <var>|N/<sub>equiv</sub>| = |Phi|+1</var>.
<p>
Every finite planar nearring can be constructed from some Ferrero pair
together with a set of orbit representatives in this way.
<p>
<strong>The problem:</strong> Find a planar nearring with 25 elements and 9 pairwise
non-equivalent multipliers.
<p>
<strong>The solution:</strong> We follow the Ferrero method described above for defining a
nearring multiplication on an additive group.
First we have to find a fixed-point-free (fpf) automorphism group of
order <var>8</var> on a group of order <var>25</var>.
<p>
We start with the cyclic group of order <var>25</var>:
First of all we ask for the existence of an fpf automorphism group
on <code>CyclicGroup(25)</code> by computing an upper bound for its order.
<p>
<pre>
gap> FpfAutomorphismGroupsMaxSize( CyclicGroup(25) );
[ 4, 1 ]
</pre>
<p>
This function returns a list with two integers, <var>4</var> and <var>1</var>.
The first number is an upper bound for the size of an fpf automorphism group;
if there is a metacyclic fpf automorphism group, then it has a cyclic normal
subgroup of index dividing the second number. These bounds are not sharp.
If the upper bound for the size of an fpf automorphism group on some group
is <var>1</var>, we know that there is no nontrivial fpf automorphism group, no
Ferrero pair, and no planar nearring on this group at all.
<p>
Here, SONATA does not exclude the possibility that the cyclic group of
order <var>25</var> has an fpf automorphism group of order <var>4</var>.
However, we can be sure that all fpf automorphism groups are cyclic and that
none of them has size <var>8</var>.
<p>
Thus we have to consider the elementary abelian group of order 25 instead.
<p>
<pre>
gap> FpfAutomorphismGroupsMaxSize( ElementaryAbelianGroup(25) );
[ 24, 2 ]
</pre>
<p>
There might even exist an fpf automorphism group of order <var>24</var>. (In fact
there is more than one. The reference manual explains how to obtain all
nearfields of size <var>25</var>.)
For our example, we could compute either a cyclic automorphism group or one
isomorphic to the quaternion group with 8 elements. Let's try the latter.
<p>
<pre>
gap> aux := FpfAutomorphismGroupsMetacyclic( [5,5], 4, -1 );
[ [ [ [ f1, f2 ] -> [ f1^2, f2^3 ], [ f1, f2 ] -> [ f2^4, f1 ] ] ],
<pc group of size 25 with 2 generators> ]
</pre>
<p>
Here, the function <code>FpfAutomorphismGroupsMetacyclic</code> determines the metacyclic
fpf automorphism groups on <code>AbelianGroup([5,5])</code> with generators <var>p,q</var>
satisfying <var>p<sup>4</sup> = 1, p<sup>q</sup> = p<sup>-1</sup></var>, and <var>q<sup>2</sup> = p<sup>2</sup></var>. For each conjugacy class
of such groups one representative is given. Conjugacy is determined within the
whole automorphism group of <code>AbelianGroup([5,5])</code>.
The actual output of the function is a list with 2 elements.
The first is not the list of fpf groups up to conjugacy but the list of
automorphisms <var>p,q</var> generating those groups.
The second element is simply the group <code>AbelianGroup([5,5])</code>, on which the
automorphisms act.
<p>
Since there is only one pair of generators <var>p,q</var>, all fpf automorphism groups
isomorphic to the quaternion group are conjugate. Now, we have our Ferrero
pair <var>(G, Phi)</var>.
<p>
<pre>
gap> phi := Group( aux[1][1] );
<group with 2 generators>
gap> G := aux[2];
<pc group of size 25 with 2 generators>
</pre>
<p>
Next we have to pick some orbit representatives.
We note that for a fixed Ferrero pair distinct choices of representatives
may yield isomorphic nearrings. The function
<code>OrbitRepresentativesForPlanarNearRing</code> returns exactly one set of
representatives of given cardinality for each isomorphism class of planar
nearrings which can be generated from <var>(G, Phi)</var>.
<p>
<pre>
gap> OrbitRepresentativesForPlanarNearRing( G, phi, 1 );
[ [ f1 ] ]
</pre>
<p>
This tells us that all planar nearrings obtained from <var>(G,Phi)</var> with one
orbit representative are in fact isomorphic.
What happens if we choose <var>2</var> representatives?
<p>
<pre>
gap> reps := OrbitRepresentativesForPlanarNearRing( G, phi, 2 );
[ [ f1, f1*f2 ], [ f1, f1^2*f2^2 ] ]
</pre>
<p>
We obtain <var>2</var> non-isomorphic planar near-rings. Let's just construct one of
them. The result will be an <code>ExplicitMultiplicationNearRing</code>.
<p>
<pre>
gap> n := PlanarNearRing( G, phi, reps[1] );
ExplicitMultiplicationNearRing ( <pc group of size 25 with
2 generators> , multiplication )
</pre>
<p>
How many non-isomorphic planar nearrings can be defined from our Ferrero pair
<var>(G,Phi)</var> in total? Since there are <var>3</var> non-trivial orbits of <var>Phi</var> on <var>G</var>,
we may choose up to <var>3</var> representatives.
<p>
<pre>
gap> Length(OrbitRepresentativesForPlanarNearRing( G, phi, 3 ));
6
</pre>
<p>
Summing all up, we get exactly <var>9</var> non-isomorphic planar nearrings with
elementary abelian additive group of order <var>25</var> whose multiplication is
defined using a quaternion group of fpf automorphisms.
<p>
<p>
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<P>
<address>SONATA-tutorial manual<br>November 2008
</address></body></html>
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