> < ^ From:

^ Subject:

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> -----------------------------------------------------------------------

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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