<html><head><title>[SONATA] 10 Nearfields, planar nearrings and weakly divisible nearrings</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP009.htm">Previous</a>] [<a href ="CHAP011.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>10 Nearfields, planar nearrings and weakly divisible nearrings</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP010.htm#SECT001">Dickson numbers</a> <li> <A HREF="CHAP010.htm#SECT002">Dickson nearfields</a> <li> <A HREF="CHAP010.htm#SECT003">Exceptional nearfields</a> <li> <A HREF="CHAP010.htm#SECT004">Planar nearrings</a> <li> <A HREF="CHAP010.htm#SECT005">Weakly divisible nearrings</a> </ol><p> <p> A <strong>nearfield</strong> is a nearring with <var>1</var> where each nonzero element has a multiplicative inverse. The (additive) group reduct of a finite nearfield is necessarily elementary abelian. For an exposition of nearfields we refer to citeWaehling:Fastkoerper. <p> 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. See citeClay:Nearrings for basic results on planar nearrings. <p> All finite nearfields are planar nearrings. <p> A left nearring <var>(N,+,cdot)</var> is called <strong>weakly divisible</strong> if <var>foralla,binN existsxinN : acdotx = b</var> or <var>bcdotx = a</var>. <p> All finite integral planar nearrings are weakly divisible. <p> <p> <h2><a name="SECT001">10.1 Dickson numbers</a></h2> <p><p> <a name = "SECT001"></a> <li><code>IsPairOfDicksonNumbers( </code><var>q</var><code>, </code><var>n</var><code> )</code> <p> A pair of Dickson numbers <var>(q,n)</var> consists of a prime power integer <var>q</var> and a natural number <var>n</var> such that for <var>p = 4</var> or <var>p</var> prime, <var>p|n</var> implies <var>p|q-1</var>. <p> <pre> gap> IsPairOfDicksonNumbers( 5, 4 ); true </pre> <p> <p> <h2><a name="SECT002">10.2 Dickson nearfields</a></h2> <p><p> <a name = "SECT002"></a> <li><code>DicksonNearFields( </code><var>q</var><code>, </code><var>n</var><code> )</code> <p> All finite nearfields with 7 exceptions can be obtained via socalled coupling maps from finite fields. These nearfields are called Dickson nearfields. <p> The multiplication map of such a Dickson nearfield is given by a pair of Dickson numbers <var>(q,n)</var> in the following way: <p> Let <var>F = GF(q<sup>n</sup>)</var> and <var>w</var> be a primitive element of <var>F</var>. Let <var>H</var> be the subgroup of <var>(Fsetminus{0},cdot)</var> generated by <var>w<sup>n</sup></var>. Then <var>{w<sup>(q^i-1)/(q-1)</sup> | 0leqileqn-1 }</var> is a set of coset representatives of <var>H</var> in <var>Fsetminus{0}</var>. For <var>finHw<sup>(q^i-1)/(q-1)</sup></var> and <var>xinF</var> define <var>f*x = fcdotx<sup>q^i</sup></var> and <var>0*x = 0</var>. Then <var>*</var> is a nearfield multiplication on the additive group <var>(F,+)</var>. <p> Note that a Dickson nearfield is not uniquely determined by <var>(q,n)</var>, since <var>w</var> can be chosen arbitrarily. Different choices of <var>w</var> may yield isomorphic nearfields. <p> <code>DicksonNearFields</code> returns a list of the non-isomorphic Dickson nearfields determined by the pair of Dickson numbers <var>(q,n)</var> <p> <pre> gap> DicksonNearFields( 5, 4 ); [ ExplicitMultiplicationNearRing ( <pc group of size 625 with 4 generators> , multiplication ), ExplicitMultiplicationNearRing ( <pc group of size 625 with 4 generators> , multiplication ) ] </pre> <p> <a name = "SECT002"></a> <li><code>NumberOfDicksonNearFields( </code><var>q</var><code>, </code><var>n</var><code> )</code> <p> <code>NumberOfDicksonNearFields</code> returns the number of non-isomorphic Dickson nearfields which can be obtained from a pair of Dickson numbers <var>(q,n)</var>. This number is given by <var>Phi(n)/k</var>. Here <var>Phi(n)</var> denotes the number of relatively prime residues modulo <var>n</var> and <var>k</var> is the multiplicative order of <var>p</var> modulo <var>n</var> where <var>p</var> is the prime divisor of <var>q</var>. <p> <pre> gap> NumberOfDicksonNearFields( 5, 4 ); 2 </pre> <p> <p> <h2><a name="SECT003">10.3 Exceptional nearfields</a></h2> <p><p> <a name = "SECT003"></a> <li><code>ExceptionalNearFields( </code><var>q</var><code> )</code> <p> There are 7 finite nearfields which cannot be obtained from finite fields via a Dickson process. They are of size <var>p<sup>2</sup></var> for <var>p = 5, 7, 11, 11, 23, 29, 59</var>. (There exist 2 exceptional nearfields of size 121.) <p> <code>ExceptionalNearFields</code> returns the list of exceptional nearfields for a given size <var>q</var>. <p> <pre> gap> ExceptionalNearFields( 25 ); [ ExplicitMultiplicationNearRing ( <pc group of size 25 with 2 generators> , multiplication ) ] </pre> <p> <a name = "SECT003"></a> <li><code>AllExceptionalNearFields()</code> <p> There are 7 finite nearfields which cannot be obtained from finite fields via a Dickson process. They are of size <var>p<sup>2</sup></var> for <var>p = 5, 7, 11, 11, 23, 29, 59</var>. (There exist 2 exceptional nearfields of size 121.) <p> <code>AllExceptionalNearFields</code> without argument returns the list of exceptional nearfields. <p> <pre> gap> AllExceptionalNearFields(); [ ExplicitMultiplicationNearRing ( <pc group of size 25 with 2 generators> , multiplication ), ExplicitMultiplicationNearRing ( <pc group of size 49 with 2 generators> , multiplication ), ExplicitMultiplicationNearRing ( <pc group of size 121 with 2 generators> , multiplication ), ExplicitMultiplicationNearRing ( <pc group of size 121 with 2 generators> , multiplication ), ExplicitMultiplicationNearRing ( <pc group of size 529 with 2 generators> , multiplication ), ExplicitMultiplicationNearRing ( <pc group of size 841 with 2 generators> , multiplication ), ExplicitMultiplicationNearRing ( <pc group of size 3481 with 2 generators> , multiplication ) ] </pre> <p> <p> <h2><a name="SECT004">10.4 Planar nearrings</a></h2> <p><p> <a name = "SECT004"></a> <li><code>PlanarNearRing( </code><var>G</var><code>, </code><var>phi</var><code>, </code><var>reps</var><code> )</code> <p> A finite <strong>Ferrero pair</strong> is a pair of groups <var>(N,Phi)</var> where <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, citeClay:Nearrings: 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>a * x = 0</var> for <var>ainNsetminusC</var>, and <var>a * x=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,+,*)</var> is a (left) planar nearring. <p> Every finite planar nearring can be constructed from some Ferrero pair together with a set of orbit representatives in this way. <p> <code>PlanarNearRing</code> returns the planar nearring on the group <var>G</var> determined by the fixed-point-free automorphism group <var>phi</var> and the list of chosen orbit representatives <var>reps</var>. <p> <pre> gap> C7 := CyclicGroup( 7 );; gap> i := GroupHomomorphismByFunction( C7, C7, x -> x^-1 );; gap> phi := Group( i );; gap> orbs := Orbits( phi, C7 ); [ [ <identity> of ... ], [ f1, f1^6 ], [ f1^2, f1^5 ], [ f1^3, f1^4 ] ] gap> # choose reps from the orbits gap> reps := [orbs[2][1], orbs[3][2]]; [ f1, f1^5 ] gap> n := PlanarNearRing( C7, phi, reps ); ExplicitMultiplicationNearRing ( <pc group of size 7 with 1 generators> , multiplication ) </pre> <p> <a name = "SECT004"></a> <li><code>OrbitRepresentativesForPlanarNearRing( </code><var>G</var><code>, </code><var>phi</var><code>, </code><var>i</var><code> )</code> <p> Let <var>(N,Phi)</var> be a Ferrero pair, and let <var>E = { e<sub>1</sub>,...,e<sub>s</sub> }</var> and <var>F = { f<sub>1</sub>,...,f<sub>t</sub> }</var> be two sets of non-zero orbit representatives. The nearring obtained from <var>N,Phi, E</var> by the Ferrero construction (see <code>PlanarNearRing</code>) is isomorphic to the nearring obtained from <var>N,Phi, F</var> iff there exists an automorphism <var>alpha</var> of <var>(N,+)</var> that normalizes <var>Phi</var> such that <var>{ alpha(e<sub>1</sub>),...,alpha(e<sub>s</sub>) } = { f<sub>1</sub>,...,f<sub>t</sub> }</var>. <p> The function <code>OrbitRepresentativesForPlanarNearRing</code> returns precisely one set of representatives of cardinality <var>i</var> for each isomorphism class of planar nearrings which can be generated from the Ferrero pair ( <var>G</var>, <var>phi</var> ). <p> <pre> gap> C7 := CyclicGroup( 7 );; gap> i := GroupHomomorphismByFunction( C7, C7, x -> x^-1 );; gap> phi := Group( i );; gap> reps := OrbitRepresentativesForPlanarNearRing( C7, phi, 2 ); [ [ f1, f1^2 ], [ f1, f1^5 ] ] gap> n1 := PlanarNearRing( C7, phi, reps[1] );; gap> n2 := PlanarNearRing( C7, phi, reps[2] );; gap> IsIsomorphicNearRing( n1, n2 ); false </pre> <p> <p> <h2><a name="SECT005">10.5 Weakly divisible nearrings</a></h2> <p><p> <a name = "SECT005"></a> <li><code>WdNearRing( </code><var>G</var><code>, </code><var>psi</var><code>, </code><var>phi</var><code>, </code><var>reps</var><code> )</code> <p> Every finite (left) weakly divisible nearring <var>(N,+,cdot)</var> can be constructed in the following way: <p> (1) Let <var>psi</var> be an endomorphism of the group <var>(N,+)</var> such that Ker <var>psi=</var> Image <var>psi<sup>r-1</sup></var> for some integer <var>r, r>0</var>. (Let <var>psi<sup>0</sup> :=</var> id.) <p> (2) Let <var>Phi</var> be an automorphism group of <var>(N,+)</var> such that <var>psiPhisubseteqPhipsi</var> and <var>Phi</var> acts fixed-point-free on <var>Nsetminus</var> Image <var>psi</var>. (That is, for each <var>varphiinPhi</var> there exists <var>varphi'inPhi</var> such that <var>psivarphi= varphi'psi</var> and for all <var>ninNsetminus</var> Image <var>psi</var> the equality <var>n^varphi= n</var> implies <var>varphi=</var> id. Note that our functions operate from the right just like GAP-mappings do.) <p> (3) Let <var>EsubseteqN</var> be a complete set of orbit representatives for <var>Phi</var> on <var>Nsetminus</var> Image <var>psi</var>, such that for all <var>e<sub>1</sub>, e<sub>2</sub>inE</var>, for all <var>varphiinPhi</var> and for all <var>1 leqi leqr-1</var> the equality <var>e<sub>1</sub><sup>varphipsi^i</sup> = e<sub>2</sub><sup>psi^i</sup></var> implies <var>varphipsi<sup>i</sup> = psi<sup>i</sup></var>. <p> Then for all <var>ninN, nneq0</var>, there are <var>igeq0 ,varphiinPhi</var> and <var>einE</var> such that <var>n = e<sup>varphipsi^i</sup></var>; furthermore, for fixed <var>n</var>, the endomorphism <var>varphipsi<sup>i</sup></var> is independent of the choice of <var>e</var> and <var>varphi</var> in the representation of <var>n</var>. <p> For all <var>xinN, einE,varphiinPhi</var> and <var>igeq0</var> define <var>0cdotx := 0</var> and <p><var> e<sup>varphipsi^i</sup>cdotx := x<sup>varphipsi^i</sup> <p></var> Then <var>(N,+,cdot)</var> is a zerosymmetric (left) wd nearring. <p> <code>WdNearRing</code> returns the wd nearring on the group <var>G</var> as defined above by the nilpotent endomorphism <var>psi</var>, the automorphism group <var>phi</var> and a list of orbit representatives <var>reps</var> where the arguments fulfill the conditions (1) to (3). <p> <pre> gap> C9 := CyclicGroup( 9 );; gap> psi := GroupHomomorphismByFunction( C9, C9, x -> x^3 );; gap> Image( psi ); Group([ f2, <identity> of ... ]) gap> Image( psi ) = Kernel( psi ); true gap> a := GroupHomomorphismByFunction( C9, C9, x -> x^4 );; gap> phi := Group( a );; gap> Size( phi ); 3 gap> orbs := Orbits( phi, C9 ); [ [ <identity> of ... ], [ f2 ], [ f2^2 ], [ f1, f1*f2, f1*f2^2 ], [ f1^2, f1^2*f2^2, f1^2*f2 ] ] gap> # choose reps from the orbits outside of Image( psi ) gap> reps := [orbs[4][1], orbs[5][1]]; [ f1, f1^2 ] gap> n := WdNearRing( C9, psi, phi, reps ); ExplicitMultiplicationNearRing ( <pc group of size 9 with 2 generators> , multiplication ) </pre> <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP009.htm">Previous</a>] [<a href ="CHAP011.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>SONATA manual<br>November 2008 </address></body></html>