<html><head><title>[SONATA-tutorial] 5 Some interesting nearrings</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href ="CHAP006.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>5 Some interesting nearrings</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP005.htm#SECT001">Nearrings generated by endomorphisms on a group</a> <li> <A HREF="CHAP005.htm#SECT002">More information than just the size</a> <li> <A HREF="CHAP005.htm#SECT003">Centralizer nearrings</a> <li> <A HREF="CHAP005.htm#SECT004">Finding affine complete groups</a> </ol><p> <p> One motivation for creating SONATA was to study particular near-rings associated with a given group <var>G</var>: the <strong>inner automorphism nearring</strong> <var>I(G)</var>, the <strong>automorphism nearring</strong> <var>A(G)</var>, and the <strong>endomorphism nearring</strong> <var>E(G)</var>. The nearring <var>I(G)</var> is the smallest subnearring of the nearring <var>M(G)</var> of all mappings from <var>G</var> into <var>G</var> that contains all inner automorphisms; similarly <var>A(G)</var> and <var>E(G)</var> are defined. citemeldrum85:NATLWG contains a lot of information on these near-rings. <p> <p> <h2><a name="SECT001">5.1 Nearrings generated by endomorphisms on a group</a></h2> <p><p> Let us compute the nearring <var>I(A<sub>4</sub>)</var>, which is the nearring of all zero-symmetric polynomial functions on the group <var>A<sub>4</sub></var>. <pre> gap> I := InnerAutomorphismNearRing ( AlternatingGroup ( 4 ) ); InnerAutomorphismNearRing( Alt( [ 1 .. 4 ] ) ) gap> Size (I); 3072 </pre> <p> For a polynomial function, we can ask for a polynomial that induces it. <p> <pre> gap> p := Random( I ); <mapping: AlternatingGroup( [ 1 .. 4 ] ) -> AlternatingGroup( [ 1 .. 4 ] ) > gap> PrintAsTerm( p ); - g1 + g2 - x - g2 + g1 + g2 + g1 - x + g2 - x + 2 * g1 - 3 * x - g1 + x + g2 - x - g2 + g1 + x - g1 + x - g1 + x + g1 + x - g2 - x + g2 - g1 - x + g1 + x gap> GeneratorsOfGroup( AlternatingGroup( 4 ) ); [ (1,2,3), (2,3,4) ] </pre> <p> We get a polynomial (not necessarily the shortest possible polynomial) that induces the polynomial function. The expressions <code>g1</code> and <code>g2</code> stand for the first and second generator of the group respectively. <p> Now we compute the nearring that is additively generated by the automorphisms of the dihedral group of order 8. This nearring is usually called <var>A (D<sub>8</sub>)</var>. <pre> gap> A := AutomorphismNearRing ( DihedralGroup ( 8 ) ); AutomorphismNearRing( <pc group of size 8 with 3 generators> ) gap> Size (A); 32 </pre> <p> Much attention has been devoted to the nearring <var>E (S<sub>4</sub>)</var>, which is the nearring additively generated by the endomorphisms on the symmetric group on four letters. <pre> gap> EndS4 := EndomorphismNearRing ( SymmetricGroup ( 4 ) ); EndomorphismNearRing( Sym( [ 1 .. 4 ] ) ) gap> Size ( EndS4 ); 927712935936 gap> F1 := last;; gap> Collected ( Factors( F1 )); [ [ 2, 35 ], [ 3, 3 ] ] </pre> In the last example, we have computed the size of <var>E (S<sub>4</sub>)</var> as <var>2<sup>35</sup> cdot3<sup>3</sup></var>. <p> We have also included some less popular examples of nearrings. One of those is the nearring <var>H (G, U)</var>. This is the nearring that is generated by all endomorphisms on <var>G</var> whose range lies in the subgroup <var>U</var> of <var>G</var>. We do an example on the group <var>16/8</var> in the classification of Thomas and Wood. It is a subdirectly irreducible group of order 16, and the factor modulo the monolith is isomorphic to the elementary abelian group of order 8. <pre> gap> G := GTW16_8; 16/8 gap> U := First ( NormalSubgroups( G ), N -> Size(N) = 2 ); Group([ ( 1, 5)( 2,10)( 3,11)( 4,12)( 6,15)( 7,16)( 8, 9)(13,14) ]) gap> HGU := RestrictedEndomorphismNearRing (G, U); RestrictedEndomorphismNearRing( 16/8, Group( [ ( 1, 5)( 2,10)( 3,11)( 4,12)( 6,15)( 7,16)( 8, 9)(13,14) ]) ) gap> Size (HGU); 8 </pre> It is interesting to compare this nearring to the nearring of all functions <var>e</var> in the endomorphism nearring <var>E (G)</var> with the property <var>e (G) subseteqU</var>. <pre> gap> EofG := EndomorphismNearRing ( G ); EndomorphismNearRing( 16/8 ) gap> EGU := NoetherianQuotient ( EofG, U, G ); NoetherianQuotient( Group( [ ( 1, 5)( 2,10)( 3,11)( 4,12)( 6,15)( 7,16)( 8, 9)(13,14) ]) ,16/8 ) gap> Size ( EGU ); 128 </pre> If <var>N</var> is a transformation nearring on <var>G</var>, and <var>U, V</var> are subsets of <var>G</var> then <code>NoetherianQuotient (N,U,V)</code> returns the collection of all mappings <var>f inN</var> such that <var>f(V) subseteqU</var>. <p> <p> <h2><a name="SECT002">5.2 More information than just the size</a></h2> <p><p> In this section, we use SONATA to produce some interesting information about the nearring <var>I(S<sub>3</sub>)</var>, which is the nearring of all zero-symmetric polynomial functions on the group <var>S<sub>3</sub></var>. <p> <pre> gap> G := SymmetricGroup ( 3 ); Sym( [ 1 .. 3 ] ) gap> I := InnerAutomorphismNearRing ( G ); InnerAutomorphismNearRing( Sym( [ 1 .. 3 ] ) ) gap> Size( I ); 54 </pre> <p> Now we would like to see how many of these 54 functions are idempotent. First a complicated version. <pre> gap> Filtered ( I, > t -> ForAll( G, g -> Image(t, g) = Image(t, Image(t, g)) ) );; gap> Length( last ); 18 </pre> Now a simpler version. <pre> gap> Filtered ( I, i -> i^2 = i );; gap> Length( last ); 18 </pre> <p> <p> <h2><a name="SECT003">5.3 Centralizer nearrings</a></h2> <p><p> Let <var>Phi</var> be a subset of the endomorphisms of a group <var>G</var>. Then we define <var>M<sub>Phi</sub> (G)</var> as the set of all mappings <var>m : G toG</var> that satisfy <var>m circvarphi= varphicircm</var> for all <var>varphiinPhi</var>. This set is closed under addition and composition of mappings, and hence a subnearring of <var>M(G)</var>. The set <var>M<sub>Phi</sub> (G)</var> is called the centralizer nearring of <var>G</var> determined by <var>Phi</var>. It need not necessarily be zero-symmetric. <p> In the following examples, we compute the centralizer nearring <var>M<sub>End (S_3)</sub> (S<sub>3</sub>)</var>. <pre> gap> G := SymmetricGroup( 3 ); Sym( [ 1 .. 3 ] ) gap> endos := Endomorphisms( G ); [ [ (1,2,3), (1,2) ] -> [ (), () ], [ (1,2,3), (1,2) ] -> [ (), (1,3) ], [ (1,2,3), (1,2) ] -> [ (), (2,3) ], [ (1,2,3), (1,2) ] -> [ (), (1,2) ], [ (1,2,3), (1,2) ] -> [ (1,2,3), (1,3) ], [ (1,2,3), (1,2) ] -> [ (1,3,2), (1,2) ], [ (1,2,3), (1,2) ] -> [ (1,3,2), (1,3) ], [ (1,2,3), (1,2) ] -> [ (1,2,3), (2,3) ], [ (1,2,3), (1,2) ] -> [ (1,2,3), (1,2) ], [ (1,2,3), (1,2) ] -> [ (1,3,2), (2,3) ] ] gap> C := CentralizerNearRing( G, endos ); CentralizerNearRing( Sym( [ 1 .. 3 ] ), ... ) gap> Size ( C ); 6 </pre> <p> An <strong>ideal</strong> of a nearring <var>(N,+,*)</var> is a subset <var>I</var> such that <var>I</var> is a normal subgroup of <var>(N,+)</var>, and for all <var>i inI</var>, <var>n,m inN</var>, we have <var>(m+i)*n - m*n inI</var> and <var>n*i inI</var>. Ideals are in one-to-one correspondence to the congruence relations on <var>(N,+,*)</var>. <p> Do you think that this nearring is simple? Alan Cannon does not think so, and, in fact, SONATA tells us: <pre> gap> I := NearRingIdeals( C ); [ < nearring ideal >, < nearring ideal >, < nearring ideal >, < nearring ideal > ] gap> List( I, Size ); [ 1, 2, 3, 6 ] </pre> So, we have ideals of size 1,2,3 and 6. <p> <p> <h2><a name="SECT004">5.4 Finding affine complete groups</a></h2> <p><p> We shall now construct all compatible (= congruence preserving) functions on the group 16/6 (Thomas-Wood-notation); this is the <var>6<sup>th</sup></var> group of order <var>16</var> in citethomaswood80:GT. It is the direct product of <var>D<sub>8</sub></var> and <var>C<sub>2</sub></var>. Let <var>G</var> be this group. We first construct the nearring <var>P(G)</var> of all polynomial functions. Then we construct all those functions that can be interpolated at every subset of <var>G</var> with at most two elements by a function in <var>P(G)</var> by using the function <code>LocalInterpolationNearRing</code>: these are the compatible functions on <var>G</var> (see citePilz:Nearrings). <pre> gap> P := PolynomialNearRing( GTW16_6 ); PolynomialNearRing( 16/6 ) gap> Size( P ); 256 gap> C := LocalInterpolationNearRing(P, 2); LocalInterpolationNearRing( PolynomialNearRing( 16/6 ), 2 ) gap> Size (C); 256 </pre> Hence the group <var>16/6</var> is <var>1</var>-affine complete. A much faster algorithm for computing the nearring of compatible functions can be used. <pre> gap> C := CompatibleFunctionNearRing( GTW16_6 ); < transformation nearring with 7 generators > gap> Size(C); 256; </pre> Finally, the fastest way to decide 1-affine completeness is to use the function <code>Is1AffineComplete</code>. <pre> gap> Is1AffineComplete( GTW16_6 ); true </pre> <p> When studying polynomial functions on direct products of groups, it is important to know the smallest positive number <var>l</var> such that the zero-function can be expressed by a term <var>a<sub>1</sub> + e<sub>1</sub>.x + a<sub>2</sub> + cdots+ e<sub>n</sub>.x + a<sub>n+1</sub></var> with <var>sume<sub>i</sub> = l</var>. This <var>l</var> has been called the <strong>length</strong> of the group by S.D.Scott. <p> <pre> gap> ScottLength( SymmetricGroup( 3 ) ); 2 </pre> <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href ="CHAP006.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>SONATA-tutorial manual<br>November 2008 </address></body></html>