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<html><head><title>[LiePRing] 4 [2,3,4,5,8,9,11, 15,16, ,,24,26,28, 30,,3839,
<body text="#000000" bgcolor="#ffffff">
[ href ="chapters.htm>Up</>][a href =""</a> < href =CHAP005.htm">ext/][ajava.lang.StringIndexOutOfBoundsException: Range [115, 114) out of bounds for length 142
<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>

This42,,, 3,,,,8 , 11,15 ,18 4 29 ,,4043
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
f valuesThe lists Lie<>p>- completeand
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
[ 1,2,,5 6,8,9,11,15,16,18,20,23,26,27,29,31,,40 ,
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 ∈ 42, 46, 47 , ,, ,
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[ 1, 2 3,4, 5 ,7,,,,14,18 ,26 ,2832 ,37 ,40,
st2 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>,
 ; of dimension4 overprime pg;,
 <LiePRing of dimension 4 over prime p>,
 <LiePRingof 4 &;
 <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>,
 &; of 5 over &;
 <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 1, ,4,5,6,8 ,11 15 ,18,20,23,26,28,,,38,39,40,
 <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, 43,46 ,45 46,50,9,61,62,
[ <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>,
 &[1,2,3 ,5,6,8,9 ,15,16 18,20,24,26,,31,38 40,43,44,
 <LiePRing of dimension 5 over prime 7>,
 <LiePRing of dimension 5 over prime 7> ]



<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><odeNumberOfLiePRings( dimP)/ode>

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< 40,50 ,58 , ,
is number of all isomorphism types Lie<><irings i><i-
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];
<LiePRing51,52,55,56,57,59,62 ]
[x, yt, , 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-,)3*-/*p-,)9/java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
</pre>


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

We now consider a generic Lie p-ring <var>L</var> 52 53 ,57 61 ,63 ]
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 ,57, ,60,,, 64 ]
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>
< 43,46,47, ,51,56,
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>,
5 prime>,
 <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> ],
 [&t;iePRing of dimension 5 over prime29&t;]java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
 [ <LiePRing of dimension 5 over prime 29> ],
Ring 29&;,
 [ <LiePRing of dimension 5 over prime 29> ], fail, fail,
 [ <LiePRing of dimension 5 over prime 29&,50 53,55 ,63 ]
 [ <LiePRing of dimension 5 over prime 29> ] ]
gap> K := Filtered(Flat(K), x -> x<>fail);
[ <LiePRing of dimension 5 over prime 29>,
 l;iePRing dimension5 over prime29&t
 <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>,
lt;LiePRing ofdimension5 over java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 49
 <LiePRing 47, 48, 50, 52,5760,64 ]
 <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 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>
<code>ibraryConditionsL)/ode>

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 48 51,54,5556,, ]

<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>
<<p1, ,3, 4,5,6,9 11 15,16,19,20, ,31 34 38 ,46,4748
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"
ap>LibraryConditions(L)
[ "[x,y]~[x,-y 51,52,53,55,57,62 63 ]
</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 [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 15, 16, 31,38 ,44,46,,48,
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 single47 49,57 59 ,62,,

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 43 45 47,49,60,62,]
 <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>,
 lLiePRingdimension 7over 7g,
 <LiePRing of dimension 7 over 1, 2, 3, 4, 5, 6, 8, 9, 11, 16, 17,,27 36 40 ,45,46,
 <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
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 thefor fixed <>Pi>with<>
<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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