> < ^ Date: Sun, 13 Aug 2000 19:36:09 +0100
> ^ From: Roger Beresford <rogerberesford@supanet.com >
^ Subject: Small Group Cayley Tables.

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Small Group Cayley Tables needed. I am a retired engineer using =
Mathematica 4 to investigate the factors of the determinants of group =
Cayley Tables, with size 2^i *3^j up to 72 elements. As my knowledge of =
group theory is limited (I am working through "A Course in Group =
Theory", J.F.Humphreys, Oxford Sci.Pubs.'96), I have not managed to =
obtain the required information from e.g. gap> SmallGroup(16,n);. every =
value of n (up to 14) elicits the unhelpful response <pc group of size =
16 with four generators>.
1. How can I give the elements of a group the names {a1,..an} or =
preferably {1..n} and then obtain the nxn table of products? In other =
words, I need a function "indexTable(group)" such that gap> =
indexTable(SmallGroup(3,1)); gives the result [[1,2,3],[2,3,1],[3,1,2]].
The point of this is that such factors ("eigenfactors" insofar as they =
are eigenvalues that have not been factorised right down to complex =
linear eigenvalues) are conserved properties in "renormalizing algebras" =
that give meaningful finite results on "division-by-zero" over a =
"non-negative number field". If one or more factors are zero, =
multiplication (including multiplication by the easily defined =
multiplicative inverse) constrains the result to a sub-algebra, just as =
conic sections are obtained on constraining the distance from some plane =
to zero.
2. Where can I find formulae that give such tables? (I know that a few =
groups exist for which there is no formula; this implies that formulae =
exist for most finite groups). I have the (Mathematica) formula for =
cyclic (k=3D1), dihedral (k=3Dm-1), and quaternion (k=3Dm/2-1) groups,
cay[m_,k_]:=3DTable[Mod[i+If[EvenQ[i],j k-k+1,,j]-1,m,1],{i,m},{j,m}];
which also gives some generalised dihedral and quaternion groups. =
Unfortunately I have not been able to generalise to groups with more =
generators.
(This message was sent on July !7th but was rejected as "Mr.Miles" had =
not understood my registration application. Steve Linton picked this up =
and sent two helpful replies, which he will repeat in reply to this =
duplicate message.)

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<DIV><FONT face=3DArial size=3D2>
<DIV><FONT face=3DArial size=3D2>Small Group Cayley Tables needed. I am =
a retired=20
engineer using Mathematica 4 to investigate the factors of the =
determinants of=20
group Cayley Tables, with&nbsp;size 2^i *3^j up to 72 elements. As my =
knowledge=20
of group theory is limited (I am working through "A Course in Group =
Theory",=20
J.F.Humphreys, Oxford Sci.Pubs.'96), I have not managed to obtain the =
required=20
information from e.g. gap&gt; SmallGroup(16,<EM>n</EM>);. every value of =

<EM>n</EM> (up to 14) elicits the unhelpful </FONT><FONT face=3DArial=20
size=3D2>response &lt;pc group of size 16 with four =
generators&gt;.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2>1. How can I give the elements of a =
group the names=20
{a1,..a<EM>n</EM>} or preferably {1..<EM>n</EM>} and then obtain=20
the&nbsp;<EM>n</EM>x<EM>n</EM> table of products? In other words, I need =
a=20
function "indexTable(group)" such that gap&gt; =
indexTable(SmallGroup(3,1));=20
gives the result [[1,2,3],[2,3,1],[3,1,2]].</FONT></DIV>
<DIV><FONT face=3DArial size=3D2>The point of this is that such factors=20
("eigenfactors" insofar as they are eigenvalues that have not been =
factorised=20
right down to complex linear eigenvalues) are conserved properties in=20
"renormalizing algebras" that give meaningful finite results on=20
"division-by-zero" over a "non-negative number field". If one or more =
factors=20
are zero, multiplication (including multiplication by the easily defined =

multiplicative inverse) constrains the result to a sub-algebra, just as =
conic=20
sections are obtained on constraining the distance from some plane to=20
zero.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2>2. Where can I find formulae that give =
such tables?=20
(I know that a few groups exist for which there is no formula; this =
implies that=20
formulae exist for most finite groups). I have the (Mathematica) formula =
for=20
cyclic (k=3D1), dihedral (k=3Dm-1), and quaternion (k=3Dm/2-1) =
groups,</FONT></DIV>
<DIV><FONT face=3DArial size=3D2>cay[m_,k_]:=3DTable[Mod[i+If[EvenQ[i],j =

k-k+1,,j]-1,m,1],{i,m},{j,m}];</FONT></DIV>
<DIV><FONT face=3DArial size=3D2>which also gives some generalised =
dihedral and=20
quaternion groups. Unfortunately I have not been able to generalise to =
groups=20
with more generators.</FONT></DIV>
<DIV>(This message was sent on July !7th but was rejected as "Mr.Miles" =
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