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As Prof Neubueser has not acknowledged my person-to-person reply to his =

forum message, I hope that I may post a farewell message.

1) Thank you Messrs. Joyner, Hibbard, and Linton for helpful replies.

2) To me, a group of size of 2^7 is large, as I thought I had made clear =

by declaring an interest in "groups with size 2^i *3^j up to 72 =

elements". I can now "formulate" most of the groups of interest to me =

apart from the uninteresting proliferation of groups of size 32 and 64.=20

3) As I only need GAP as a reference atlas for Cayley Tables, I shall =

guard my version with "AsSortedList".

4) The important paper by Formanek and Sibley, (identified by David =

Joyner, GAP forum 15 Aug) follows Dedekind & Frobenius in working with =

the INVERSE of the Cayley table (van der Waarden, History of Algebra, =

p224 et seq). This eliminates the key renormalization cases where one or =

more of the Cayley determinant factors take on the value of zero. =

Incidently, Real, Complex, Quaternion and Octonion algebras are =

degenerate because their tables have only one (repeated) factor - they =

cannot renormalize. =20

4) Prof Neubueser did not understand what I said about "renormalizing =

algebras"; perhaps because I have not published anything about them. I =

had hoped that Gap might provide a pre-publication critique of my work. =

I can e-mail a demonstration GAP session to anyone showing an interest.

5) The most important renormalizing algebras have multi-phase "Polar =

Duals" that cannot be handled as GAP rationals, but are easy in =

Mathematica. E-mail discussions with other Mathematica users will be =

welcomed.

Good-bye.

rogerberesford@supanet.com

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<DIV><FONT face=3DArial size=3D2>As Prof Neubueser has not acknowledged =

my=20

person-to-person reply to his forum message, I hope that I may post a =

farewell=20

message.</FONT></DIV>

<DIV><FONT face=3DArial size=3D2>1) Thank you Messrs. Joyner, =

Hibbard</FONT><FONT=20

face=3DArial size=3D2>, and Linton for helpful replies.</DIV>

<DIV>

<DIV><FONT face=3DArial size=3D2>2) To me, a group of size of 2^7 is =

</FONT>large,=20

as I thought I had made clear by declaring an interest in "groups =

with size=20

2^i *3^j up to 72 elements". I can now "formulate" most of the groups of =

interest to me apart from the uninteresting proliferation of groups of =

size 32=20

and 64. </DIV>

<DIV>3) As I only need GAP as a reference atlas for Cayley Tables, I =

shall guard=20

my version with "AsSortedList".</DIV>

<DIV>4) The important paper by Formanek and Sibley, (identified by David =

Joyner,=20

GAP forum 15 Aug) follows Dedekind & Frobenius in working with the =

INVERSE=20

of the Cayley table (van der Waarden, History of Algebra, p224 et seq). =

This=20

eliminates the key renormalization cases where one or more of the Cayley =

determinant factors take on the value of zero. Incidently, Real, =

Complex,=20

Quaternion and Octonion algebras are degenerate because their tables =

have only=20

one (repeated) factor - they cannot renormalize. </DIV>

<DIV></FONT><FONT face=3DArial size=3D2>4) Prof Neubueser did not =

understand what I=20

said about "renormalizing algebras"; perhaps because I have not =

published=20

anything about them. I had hoped that Gap might provide a =

pre-publication=20

critique of my work. I can e-mail a demonstration GAP session to anyone =

showing=20

an interest.</FONT></DIV>

<DIV><FONT face=3DArial size=3D2>5) The most important renormalizing =

algebras have=20

multi-phase "Polar Duals" that cannot be handled as GAP rationals, but =

are easy=20

in Mathematica. E-mail discussions with other Mathematica users will be=20

welcomed.</FONT></DIV>

<DIV><FONT face=3DArial size=3D2>Good-bye.</FONT></DIV>

<DIV><FONT face=3DArial size=3D2><A=20

href=3D"mailto:rogerberesford@supanet.com">rogerberesford@supanet.com</A>=

</FONT></DIV></DIV></BODY></HTML>

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