<html><head><title>[SONATA-tutorial] 7 Planar nearrings</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP006.htm">Previous</a>] [<a href ="CHAP008.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <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> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP006.htm">Previous</a>] [<a href ="CHAP008.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>SONATA-tutorial manual<br>November 2008 </address></body></html>