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<h1>4 The Database</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP004.htm#SECT001">Accessing Lie p-rings</a>
<li> <A HREF="CHAP004.htm#SECT002">Numbers of Lie p-rings</a>
<li> <A HREF="CHAP004.htm#SECT003">Searching the database</a>
<li> <A HREF="CHAP004.htm#SECT004">More details</a>
<li> <A HREF="CHAP004.htm#SECT005">Special functions for dimension 7</a>
<li> <A HREF="CHAP004.htm#SECT006">Dimension 8 and maximal class</a>
</ol>

This package gives access to the database of Lie p-rings of order at most
p7 as determined by Mike Newman, Eamonn O'Brien and Michael Vaughan-Lee,
see <a href="biblio.htm#NOV04"><cite>NOV04</cite></a> and <a href="biblio.htm#OVL05"><cite>OVL05</cite></a>. A description of the database can also be
found in <a href="biblio.htm#Notes"><cite>Notes</cite></a>.


For each n ∈ {1, …, 7} this package contains a (finite) list of
generic presentations of Lie p-rings. For each prime p ≥ 5, each
of the generic Lie p-rings gives rise to a family of Lie p-rings over
the considered prime p by specialising the indeterminates to a certain list
of values. The resulting lists of Lie p-rings provides a complete and
irredundant set of isomorphism type representatives of the Lie p-rings of
order pn. The generic Lie p-rings of p-class at most 2 can also be
considered for the prime p=3 and yield a list of isomorphism type
representatives for the Lie p-rings of order 3n and p-class at most
2.


The Lazard correspondence has been used to check the correctness of the
database of Lie p-rings: for various small primes it has been checked
that the Lie p-rings of this database define non-isomorphic finite
p-groups.


In the following we describe functions to access the database. Throughout
this chapter, we assume that dim ∈ {1, …, 7} and P is a prime
with P ≠ 2.


<h2><a name="SECT001">4.1 Accessing Lie p-rings</a></h2>

<a name = "SSEC001.2"></a>
<li><code>LiePRingsByLibrary( dim )</code>
<li><code>LiePRingsByLibrary( dim, gen, cl )</code>

returns the generic Lie p-rings of dimension dim in the database. The
second form returns the Lie p-rings of minimal generator number gen
and p-class cl only.

<li><code>LiePRingsByLibrary( dim, P )</code>
<li><code>LiePRingsByLibrary( dim, P, gen, cl )</code>

returns isomorphism type representatives of ordinary Lie p-rings of
dimension dim for the prime P. The second form returns the Lie p-rings
of minimal generator number gen and p-class cl only. The function
assumes P ≥ 3 and for P = 3 there are only the Lie p-rings of
p-class at most 2 available.

The first example yields the generic Lie p-rings of dimension 4.

<pre>
gap> LiePRingsByLibrary(4);
[ <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p>,
 <LiePRing of dimension 4 over prime p> ]
</pre>

The next example yields the isomorphism type representatives of Lie
p-rings of dimension 3 for the prime 5.

<pre>
gap> LiePRingsByLibrary(3, 5);
[ <LiePRing of dimension 3 over prime 5>,
 <LiePRing of dimension 3 over prime 5>,
 <LiePRing of dimension 3 over prime 5>,
 <LiePRing of dimension 3 over prime 5>,
 <LiePRing of dimension 3 over prime 5> ]
</pre>

The following example extracts the generic Lie p-rings of dimension
5 with minimal generator number 2 and p-class 4.

<pre>
gap> LiePRingsByLibrary(5, 2, 4);
[ <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p> ]
</pre>

Finally, we determine the isomorphism type representatives of Lie
p-rings of dimension 5, minimal generator number 2 and p-class
4 for the prime 7.

<pre>
gap> LiePRingsByLibrary(5, 7, 2, 4);
[ <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7> ]
</pre>


<h2><a name="SECT002">4.2 Numbers of Lie p-rings</a></h2>

<a name = "SSEC002.2"></a>
<li><code>NumberOfLiePRings( dim )</code>

returns the number of generic Lie p-rings in the database of the
considered dimension for dim { 1, …, 7}.

<pre>
gap> List([1..7], x -> NumberOfLiePRings(x));
[ 1, 2, 5, 15, 75, 542, 4773 ]
</pre>

<li><code>NumberOfLiePRings( dim, P )</code>

returns the number of isomorphism types of ordinary Lie p-rings of order
Pdim in the database. If P ≥ 5, then this is the number of all
isomorphism types of Lie p-rings of order Pdim and if P = 3 then
this is the number of all isomorphism types of Lie p-rings of p-class
at most 2. If P ≥ 7, then this number coincides with
NumberSmallGroups(Pdim).

<a name = "SSEC002.3"></a>
<li><code>NumberOfLiePRingsInFamily( L )</code>

returns the number of Lie p-rings associated to L as a polynomial in
<var>p</var> and possibly some residue classes.

<pre>
gap> L := LiePRingsByLibrary(7)[780];
<LiePRing of dimension 7 over prime p with parameters
[ x, y, z, t, s, u, v ]>
gap> NumberOfLiePRingsInFamily(L);
-1/3*p^5*(p-1,3)+p^5-1/3*p^4*(p-1,3)+p^4-1/3*p^3*(p-1,3)+p^3-1/3*p^2*(p-1,3)
+p^2-p*(p-1,3)+3*p-3/2*(p-1,3)+9/2
</pre>


<h2><a name="SECT003">4.3 Searching the database</a></h2>

We now consider a generic Lie p-ring <var>L</var> from the database and consider
the family of ordinary Lie p-rings that arise from it.

<a name = "SSEC003.1"></a>
<li><code>LiePRingsInFamily( L, P )</code>

takes as input a generic Lie p-ring <var>L</var> from the database and a prime <var>P</var>
and returns all Lie p-rings determined by <var>L</var> and <var>P</var> up to isomorphism.
This function returns fail if the generic Lie p-ring does not exist for
the special prime <var>P</var>; this may be due to the conditions on the prime or
(if P=3) to the p-class of the Lie p-ring.

<pre>
gap> L := LiePRingsByLibrary(7)[118];
<LiePRing of dimension 7 over prime p with parameters [ x, y ]>
gap> LibraryConditions(L);
[ "[x,y]~[x,-y]", "p=1 mod 4" ]
gap> LiePRingsInFamily(L, 7);
fail
gap> Length(LiePRingsInFamily(L,13));
91
gap> 13^2;
169
</pre>

The following example shows how to determine all Lie p-rings of dimension
5 and p-class 4 over the prime 29 up to isomorphism.

<pre>
gap> L := LiePRingsByLibrary(5);;
gap> L := Filtered(L, x -> PClassOfLiePRing(x)=4);
[ <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p>,
 <LiePRing of dimension 5 over prime p> ]
gap> K := List(L, x-> LiePRingsInFamily(x, 29));
[ [ <LiePRing of dimension 5 over prime 29> ],
 [ <LiePRing of dimension 5 over prime 29> ],
 [ <LiePRing of dimension 5 over prime 29> ], fail, fail,
 [ <LiePRing of dimension 5 over prime 29> ],
 [ <LiePRing of dimension 5 over prime 29> ],
 [ <LiePRing of dimension 5 over prime 29> ],
 [ <LiePRing of dimension 5 over prime 29> ],
 [ <LiePRing of dimension 5 over prime 29> ],
 [ <LiePRing of dimension 5 over prime 29> ], fail, fail,
 [ <LiePRing of dimension 5 over prime 29> ],
 [ <LiePRing of dimension 5 over prime 29> ] ]
gap> K := Filtered(Flat(K), x -> x<>fail);
[ <LiePRing of dimension 5 over prime 29>,
 <LiePRing of dimension 5 over prime 29>,
 <LiePRing of dimension 5 over prime 29>,
 <LiePRing of dimension 5 over prime 29>,
 <LiePRing of dimension 5 over prime 29>,
 <LiePRing of dimension 5 over prime 29>,
 <LiePRing of dimension 5 over prime 29>,
 <LiePRing of dimension 5 over prime 29>,
 <LiePRing of dimension 5 over prime 29>,
 <LiePRing of dimension 5 over prime 29>,
 <LiePRing of dimension 5 over prime 29> ]
</pre>


<h2><a name="SECT004">4.4 More details</a></h2>

Let L be a Lie p-ring from the database. Then the following additional
attributes are available.

<a name = "SSEC004.1"></a>
<li><code>LibraryName(L)</code>

returns a string with the name of L in the database. See p567.pdf for
further background.

<a name = "SSEC004.2"></a>
<li><code>ShortPresentation(L)</code>

returns a string exhibiting a short presentation of L.

<a name = "SSEC004.3"></a>
<li><code>LibraryConditions(L)</code>

returns the conditions on L. This is a list of two strings. The first
string exhibits the conditions on the parameters of L, the second shows
the conditions on primes.

<a name = "SSEC004.4"></a>
<li><code>MinimalGeneratorNumberOfLiePRing(L)</code>

returns the minimial generator number of L.

<a name = "SSEC004.5"></a>
<li><code>PClassOfLiePRing(L)</code>

returns the p-class of L.

<pre>
gap> L := LiePRingsByLibrary(7)[118];
<LiePRing of dimension 7 over prime p with parameters [ x, y ]>
gap> LibraryName(L);
"7.118"
gap> LibraryConditions(L);
[ "[x,y]~[x,-y]", "p=1 mod 4" ]
</pre>

All of the information listed in this section is inherited when L
is specialised.

<pre>
gap> L := LiePRingsByLibrary(7)[118];
<LiePRing of dimension 7 over prime p with parameters [ x, y ]>
gap> K := SpecialiseLiePRing(L, 13, ParametersOfLiePRing(L), [0,0]);
<LiePRing of dimension 7 over prime 13>
gap> LibraryName(K);
"7.118"
gap> LibraryConditions(K);
[ "[x,y]~[x,-y]", "p=1 mod 4" ]
</pre>

The following example shows how to find a Lie p-ring with a
given name in the database.

<pre>
gap> L := LiePRingsByLibrary(7);;
gap> Filtered(L, x -> LibraryName(x) = "7.1010")[1];
<LiePRing of dimension 7 over prime p>
</pre>


<h2><a name="SECT005">4.5 Special functions for dimension 7</a></h2>

The database of Lie p-rings of dimension 7 is very large and it may
be time-consuming (or even impossible due to storage problems) to generate
all Lie p-rings of dimension 7 for a given prime P.

Thus there are some special functions available that can be used to access
a particular set of Lie p-rings of dimension 7 only. In particular, it
is possible to consider the descendants of a single Lie p-ring of smaller
dimension by itself. The Lie p-rings of this type are all stored in one
file of the library. Thus, equivalently, it is possible to access the Lie
p-rings in one single file only.

The table LIE_TABLE contains a list of all possible files together with
the number of Lie p-rings generated by their corresponding Lie p-rings.

<a name = "SSEC005.2"></a>
<li><code>LiePRingsDim7ByFile( nr )</code>

returns the generic Lie p-rings in file number nr.

<li><code>LiePRingsDim7ByFile( nr, P )</code>

returns the isomorphism types of Lie p-rings in file number nr for
the prime <var>P</var>.

<pre>
gap> LIE_TABLE[100];
[ "3gen/gapdec6.139", 1/2*p+(p-1,3)+3/2 ]
gap> LiePRingsDim7ByFile(100);
[ <LiePRing of dimension 7 over prime p>,
 <LiePRing of dimension 7 over prime p>,
 <LiePRing of dimension 7 over prime p>,
 <LiePRing of dimension 7 over prime p>,
 <LiePRing of dimension 7 over prime p with parameters [ x ]> ]
gap> LiePRingsDim7ByFile(100, 7);
[ <LiePRing of dimension 7 over prime 7>,
 <LiePRing of dimension 7 over prime 7>,
 <LiePRing of dimension 7 over prime 7>,
 <LiePRing of dimension 7 over prime 7>,
 <LiePRing of dimension 7 over prime 7>,
 <LiePRing of dimension 7 over prime 7>,
 <LiePRing of dimension 7 over prime 7>,
 <LiePRing of dimension 7 over prime 7> ]
</pre>


<h2><a name="SECT006">4.6 Dimension 8 and maximal class</a></h2>

Recently, Lee and Vaughan-Lee <a href="biblio.htm#MC8"><cite>MC8</cite></a> determined the Lie p-rings
of dimension 8 with maximal class up to isomorphism. This classification
is now also available in the Lie p-ring package via the following functions.

<a name = "SSEC006.1"></a>
<li><code>LiePRingsByLibraryMC8()</code>

returns a list of 69 generic Lie p-rings. For each of these
the following function returns the isomorphism types of Lie p-rings
in the family for a fixed prime P with P ≥ 5.

<a name = "SSEC006.2"></a>
<li><code>LiePRingsInFamilyMC8(L, P)</code>


[<a href = "chapters.htm">Up</a>] [<a href ="CHAP003.htm">Previous</a>] [<a href ="CHAP005.htm">Next</a>] [<a href = "theindex.htm">Index</a>]

<address>LiePRing manual June 2024
</address></body></html>

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