<html><head><title>[SONATA-tutorial] 5 Some interesting nearrings</title></head>
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<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>
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<P>
<address>SONATA-tutorial manual<br>November 2008
</address></body></html>
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