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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 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> 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 <pc group of size 16 with four =

generators>.</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 <EM>n</EM>x<EM>n</EM> table of products? In other words, I need =

a=20

function "indexTable(group)" such that gap> =

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" =

had not=20

understood my registration application. Steve Linton picked this up and =

sent two=20

helpful replies, which he will repeat in reply to this duplicate=20

message.)</DIV></FONT></DIV></BODY></HTML>

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