Dear Mr. Beresford,

on August 13 you sent the following letter to the GAP forum.

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.)

Let me first comment that indeed Steve Linton had sent you a letter in
which he had given all help that we could provide for the use of GAP
with your problems. Therefore there was no point in sending the
identical questions to the GAP forum which means that they are sent to
hundreds of colleagues. Nevertheless, since we have the habit of
answering in the forum all questions to the forum I am now forced to
repeat here much of what Steve had already said.

As to your first question, Steve had answered:

-----------------------------------------------------------------------

It is easy enough to obtain numerical Cayley (multiplication) tables of
small enough groups using GAP by writing a GAP function such as the
following:

table := function(g)
local els;
els := AsListSorted(g);
return List(els, x-> List(els, y-> Position(els, x*y)));
end;

Here is a short sample session, where I apply this:

gap> table := function(g)
>       local els;
>       els := AsListSorted(g);
>       return List(els, x-> List(els, y-> Position(els, x*y)));
> end;
function( g ) ... end
gap> table(SmallGroup(16,5));
#I  The command `AsListSorted' will *not* be supported infurther versions!
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 ], 
  [ 2, 4, 6, 7, 8, 9, 5, 11, 12, 13, 14, 10, 15, 1, 16, 3 ], 
  [ 3, 6, 1, 9, 10, 2, 12, 13, 4, 5, 15, 7, 8, 16, 11, 14 ], 
  [ 4, 7, 9, 5, 11, 12, 8, 14, 10, 15, 1, 13, 16, 2, 3, 6 ], 
  [ 5, 8, 10, 11, 1, 13, 14, 2, 15, 3, 4, 16, 6, 7, 9, 12 ], 
  [ 6, 9, 2, 12, 13, 4, 10, 15, 7, 8, 16, 5, 11, 3, 14, 1 ], 
  [ 7, 5, 12, 8, 14, 10, 11, 1, 13, 16, 2, 15, 3, 4, 6, 9 ], 
  [ 8, 11, 13, 14, 2, 15, 1, 4, 16, 6, 7, 3, 9, 5, 12, 10 ], 
  [ 9, 12, 4, 10, 15, 7, 13, 16, 5, 11, 3, 8, 14, 6, 1, 2 ], 
  [ 10, 13, 5, 15, 3, 8, 16, 6, 11, 1, 9, 14, 2, 12, 4, 7 ], 
  [ 11, 14, 15, 1, 4, 16, 2, 7, 3, 9, 5, 6, 12, 8, 10, 13 ], 
  [ 12, 10, 7, 13, 16, 5, 15, 3, 8, 14, 6, 11, 1, 9, 2, 4 ], 
  [ 13, 15, 8, 16, 6, 11, 3, 9, 14, 2, 12, 1, 4, 10, 7, 5 ], 
  [ 14, 1, 16, 2, 7, 3, 4, 5, 6, 12, 8, 9, 10, 11, 13, 15 ], 
  [ 15, 16, 11, 3, 9, 14, 6, 12, 1, 4, 10, 2, 7, 13, 5, 8 ], 
  [ 16, 3, 14, 6, 12, 1, 9, 10, 2, 7, 13, 4, 5, 15, 8, 11 ] ]
gap> 
gap> CharacteristicPolynomial(last);
-504862081024000*x_1-37307822899200*x_1^2+2592731493498880*x_1^3+
191637036826624*x_1^4-405882688387072*x_1^5-30208268546560*x_1^6+
11900727945984*x_1^7+913245167680*x_1^8-86985545728*x_1^9-7446051840*x_1^10+
79377856*x_1^11+14051152*x_1^12+81632*x_1^13-8440*x_1^14-84*x_1^15+x_1^16
gap> Factors(last);
[ -136+x_1, x_1, 20+x_1, 32+x_1, -800+x_1^2, -8+x_1^2, 
  113288-1160*x_1^2+x_1^4, 8-40*x_1^2+x_1^4 ]
gap> 

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Let me in addition to what Steve has already explained say a word to
your remark:

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>.

GAP's answer just gives you some basic information about the group (that
it is of size 16) and the way it is represented in GAP, namely by a
polycyclic presentation (this is what pc stands for) on four generators.
As the sample session given by Steve shows, you can directly apply all
GAP functions to the group that you call from GAP's group libraries by
"SmallGroup(16,5);".

Neither I nor apparently Steve do understand what you are saying about
"renormalizing algebras" etc. and I doubt that other members of the GAP
forum will be able to understand it either.

As to your second question, I think you aren't sufficiently aware of
the vast number and variety of (isomorphism types of) even rather small
finite groups; e.g., it is (since very recently) known that there are
49 487 365 422 groups of order 2^10 and 423 164 062 more groups of
order at most 2000 excluding order 2^10. While the groups of order 2^10
have just been counted the others have been explicitly determined.
I think from these figures it will be clear to you that the facts are
just the contrary of what you express in your sentence "I know that a few
groups exist for which there is no formula; this implies that formulae
exist for most finite groups". It will be possible to write down such
formulae for very special classes of groups such as the cyclic, dihedral
and quaternion groups that you mention, but there are no general formulae
known and in fact will never been known for "most" finite groups.

Since, as I indicated in the beginning, this will hardly be a discussion
that interests other members of the GAP forum, would you please write
to
gap-trouble@dcs.st-and.ac.uk

if you have further questions.

With kind regards, Joachim Neubueser

---------------------------
Prof. em. J. Neubueser
Lehrstuhl D fuer Mathematik
RWTH Aachen
Germany
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