% \Chapter{Some interesting nearrings} % One motivation for creating SONATA was to study particular near-rings associated with a given group $G$: the *inner automorphism nearring* $I(G)$, the *automorphism nearring* $A(G)$, and the *endomorphism nearring* $E(G)$. The nearring $I(G)$ is the smallest subnearring of the nearring $M(G)$ of all mappings from $G$ into $G$ that contains all inner automorphisms; similarly $A(G)$ and $E(G)$ are defined. \cite{meldrum85:NATLWG} contains a lot of information on these near-rings. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Nearrings generated by endomorphisms on a group} Let us compute the nearring $I(A_4)$, which is the nearring of all zero-symmetric polynomial functions on the group $A_4$. \beginexample gap> I := InnerAutomorphismNearRing ( AlternatingGroup ( 4 ) ); InnerAutomorphismNearRing( Alt( [ 1 .. 4 ] ) ) gap> Size (I); 3072 \endexample For a polynomial function, we can ask for a polynomial that induces it. \beginexample 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) ] \endexample We get a polynomial (not necessarily the shortest possible polynomial) that induces the polynomial function. The expressions `g1' and `g2' stand for the first and second generator of the group respectively. Now we compute the nearring that is additively generated by the automorphisms of the dihedral group of order 8. This nearring is usually called $A (D_8)$. \beginexample gap> A := AutomorphismNearRing ( DihedralGroup ( 8 ) ); AutomorphismNearRing( <pc group of size 8 with 3 generators> ) gap> Size (A); 32 \endexample Much attention has been devoted to the nearring $E (S_4)$, which is the nearring additively generated by the endomorphisms on the symmetric group on four letters. \beginexample gap> EndS4 := EndomorphismNearRing ( SymmetricGroup ( 4 ) ); EndomorphismNearRing( Sym( [ 1 .. 4 ] ) ) gap> Size ( EndS4 ); 927712935936 gap> F1 := last;; gap> Collected ( Factors( F1 )); [ [ 2, 35 ], [ 3, 3 ] ] \endexample In the last example, we have computed the size of $E (S_4)$ as $2^{35} \cdot 3^3$. We have also included some less popular examples of nearrings. One of those is the nearring $H (G, U)$. This is the nearring that is generated by all endomorphisms on $G$ whose range lies in the subgroup $U$ of $G$. We do an example on the group $16/8$ 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. \beginexample 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 \endexample It is interesting to compare this nearring to the nearring of all functions $e$ in the endomorphism nearring $E (G)$ with the property $e (G) \subseteq U$. \beginexample 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 \endexample If $N$ is a transformation nearring on $G$, and $U, V$ are subsets of $G$ then `NoetherianQuotient (N,U,V)' returns the collection of all mappings $f \in N$ such that $f(V) \subseteq U$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{More information than just the size} In this section, we use SONATA to produce some interesting information about the nearring $I(S_3)$, which is the nearring of all zero-symmetric polynomial functions on the group $S_3$. \beginexample gap> G := SymmetricGroup ( 3 ); Sym( [ 1 .. 3 ] ) gap> I := InnerAutomorphismNearRing ( G ); InnerAutomorphismNearRing( Sym( [ 1 .. 3 ] ) ) gap> Size( I ); 54 \endexample Now we would like to see how many of these 54 functions are idempotent. First a complicated version. \beginexample gap> Filtered ( I, > t -> ForAll( G, g -> Image(t, g) = Image(t, Image(t, g)) ) );; gap> Length( last ); 18 \endexample Now a simpler version. \beginexample gap> Filtered ( I, i -> i^2 = i );; gap> Length( last ); 18 \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Centralizer nearrings} Let $\Phi$ be a subset of the endomorphisms of a group $G$. Then we define $M_{\Phi} (G)$ as the set of all mappings $m : G \to G$ that satisfy $m \circ \varphi = \varphi \circ m$ for all $\varphi \in \Phi$. This set is closed under addition and composition of mappings, and hence a subnearring of $M(G)$. The set $M_{\Phi} (G)$ is called the centralizer nearring of $G$ determined by $\Phi$. It need not necessarily be zero-symmetric. In the following examples, we compute the centralizer nearring $M_{End (S_3)} (S_3)$. \beginexample 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 \endexample An *ideal* of a nearring $(N,+,*)$ is a subset $I$ such that $I$ is a normal subgroup of $(N,+)$, and for all $i \in I$, $n,m \in N$, we have $(m+i)*n - m*n \in I$ and $n*i \in I$. Ideals are in one-to-one correspondence to the congruence relations on $(N,+,*)$. Do you think that this nearring is simple? Alan Cannon does not think so, and, in fact, SONATA tells us: \beginexample gap> I := NearRingIdeals( C ); [ < nearring ideal >, < nearring ideal >, < nearring ideal >, < nearring ideal > ] gap> List( I, Size ); [ 1, 2, 3, 6 ] \endexample So, we have ideals of size 1,2,3 and 6. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Finding affine complete groups} We shall now construct all compatible (= congruence preserving) functions on the group 16/6 (Thomas-Wood-notation); this is the $6^{th}$ group of order $16$ in \cite{thomaswood80:GT}. It is the direct product of $D_8$ and $C_2$. Let $G$ be this group. We first construct the nearring $P(G)$ of all polynomial functions. Then we construct all those functions that can be interpolated at every subset of $G$ with at most two elements by a function in $P(G)$ by using the function `LocalInterpolationNearRing': these are the compatible functions on $G$ (see \cite{Pilz:Nearrings}). \beginexample 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 \endexample Hence the group $16/6$ is $1$-affine complete. A much faster algorithm for computing the nearring of compatible functions can be used. \beginexample gap> C := CompatibleFunctionNearRing( GTW16_6 ); < transformation nearring with 7 generators > gap> Size(C); 256; \endexample Finally, the fastest way to decide 1-affine completeness is to use the function `Is1AffineComplete'. \beginexample gap> Is1AffineComplete( GTW16_6 ); true \endexample When studying polynomial functions on direct products of groups, it is important to know the smallest positive number $l$ such that the zero-function can be expressed by a term $a_1 + e_1.x + a_2 + \cdots + e_n.x + a_{n+1}$ with $\sum e_i = l$. This $l$ has been called the *length* of the group by S.D.Scott. \beginexample gap> ScottLength( SymmetricGroup( 3 ) ); 2 \endexample %%% Local Variables: %%% mode: latex %%% TeX-master: "manual" %%% End: