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. <[>meldrum85:NATLWG] contains a lot of information on these near-rings.
Let us compute the nearring I(A4), which is the nearring of all zero-symmetric polynomial functions on the group A4.
gap> I := InnerAutomorphismNearRing ( AlternatingGroup ( 4 ) ); InnerAutomorphismNearRing( Alt( [ 1 .. 4 ] ) ) gap> Size (I); 3072
For a polynomial function, we can ask for a polynomial that induces it.
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) ]
We get a polynomial (not necessarily the shortest possible polynomial) that induces
the polynomial function. The expressions
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 (D8).
gap> A := AutomorphismNearRing ( DihedralGroup ( 8 ) ); AutomorphismNearRing( <pc group of size 8 with 3 generators> ) gap> Size (A); 32
Much attention has been devoted to the nearring E (S4), which is the nearring additively generated by the endomorphisms on the symmetric group on four letters.
gap> EndS4 := EndomorphismNearRing ( SymmetricGroup ( 4 ) ); EndomorphismNearRing( Sym( [ 1 .. 4 ] ) ) gap> Size ( EndS4 ); 927712935936 gap> F1 := last;; gap> Collected ( Factors( F1 )); [ [ 2, 35 ], [ 3, 3 ] ]In the last example, we have computed the size of E (S4) as 235 cdot33.
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.
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); 8It is interesting to compare this nearring to the nearring of all functions e in the endomorphism nearring E (G) with the property e (G) subseteqU.
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 ); 128If 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 inN such that f(V) subseteqU.
In this section, we use SONATA to produce some interesting information about the nearring I(S3), which is the nearring of all zero-symmetric polynomial functions on the group S3.
gap> G := SymmetricGroup ( 3 ); Sym( [ 1 .. 3 ] ) gap> I := InnerAutomorphismNearRing ( G ); InnerAutomorphismNearRing( Sym( [ 1 .. 3 ] ) ) gap> Size( I ); 54
Now we would like to see how many of these 54 functions are idempotent. First a complicated version.
gap> Filtered ( I, > t -> ForAll( G, g -> Image(t, g) = Image(t, Image(t, g)) ) );; gap> Length( last ); 18Now a simpler version.
gap> Filtered ( I, i -> i^2 = i );; gap> Length( last ); 18
Let Phi be a subset of the endomorphisms of a group G. Then we define MPhi (G) as the set of all mappings m : G toG that satisfy m circvarphi= varphicircm for all varphiinPhi. This set is closed under addition and composition of mappings, and hence a subnearring of M(G). The set MPhi (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 MEnd (S_3) (S3).
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
An ideal of a nearring (N,+,*) is a subset I such that I is a normal subgroup of (N,+), and for all i inI, n,m inN, we have (m+i)*n - m*n inI and n*i inI. 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:
gap> I := NearRingIdeals( C ); [ < nearring ideal >, < nearring ideal >, < nearring ideal >, < nearring ideal > ] gap> List( I, Size ); [ 1, 2, 3, 6 ]So, we have ideals of size 1,2,3 and 6.
We shall now construct all compatible (= congruence preserving) functions
on the group 16/6 (Thomas-Wood-notation); this is the 6th group
of order 16 in <[>thomaswood80:GT].
It is the direct
product of D8 and C2. 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
these are the compatible functions on G (see <[>Pilz:Nearrings)].
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); 256Hence the group 16/6 is 1-affine complete. A much faster algorithm for computing the nearring of compatible functions can be used.
gap> C := CompatibleFunctionNearRing( GTW16_6 ); < transformation nearring with 7 generators > gap> Size(C); 256;Finally, the fastest way to decide 1-affine completeness is to use the function
gap> Is1AffineComplete( GTW16_6 ); true
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 a1 + e1.x + a2 + cdots+ en.x + an+1 with sumei = l. This l has been called the length of the group by S.D.Scott.
gap> ScottLength( SymmetricGroup( 3 ) ); 2
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