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(1) How can I extract the defining relationships for a group from GAP? =

Using gmmnn as shorthand for SmallGroup(mm,nn), I particularly need them =

for g2003, g2704, fifteen of the 32-element groups, g3603, & g3609. I =

have written a Mathematica package that handles small groups and loops =

(up to 120 elements without difficulty), as Cayley tables. Any isomorph =

of a group with less than 64 elements (and their subgroups) can be =

identified. Other properties, compositions, etc. are handled. Most of =

these groups can be created (as Cayley tables) by this package, (e.g. =

SL(2,5) from the permutations given in the GAP Tutorial, section 6.2) =

but I have failed to find relationships to create those mentioned.

(2) My internet searches have failed to find anything on the following =

subject, so I am taking the liberty of "asking the experts" via GAP.

Is there a literature and established nomenclature for "Signed Tables" =

(?hypogroups, grouplets, groups "up-to-a-sign"?). These have Cayley =

tables with signed products. The multiplication tables for many algebras =

are signed tables - their structure constants are signed single =

elements. Many such algebras create group tables on composition with =

some groups (allowing for signs); they are "collapsed" from groups via =

equivalence relationships, an (m)x(m) group table becoming an =

(m/n)x(n/n) algebra. I am cataloguing them in an extension of the GAP =

Group Atlas.

Examples of signed table algebras, with my identifiers:-

Quaternion Pauli sigma Davenport Clifford(2) g0405 g0410 g0408 g0409 1 2 3 4 1 2 3 4 1 2 3 4 1 2 = 3 4 2 -1 4 -3 2 1 -4i 3i 2 -1 4 -3 2 1 = 4 3 3 -4 -1 2 3 4i 1 -2i 3 4 -1 -2 3 -4 = 1 -2 4 3 -2 -1 4 -3i 2i 1 4 -3 -2 1 4 -3 = 2 -1

Using ":+" for composition allowing for signs, these compose as =

follows:- g0405:+g0201->g0805; g0410:+g0402->g1613; g0408:+g0201->g0802; =

g0409:+g0402->g0803. I am studying the algebras that employ conservative =

(in Frobenius's sense i.e all groups plus octonions) Moufang loops and =

signed tables as multiplication tables. They have many interesting =

properties (including, pace Dr Neuebuser) renormalisation as constrained =

sub-algebras. They provide a general framework for many algebras, as =

evinced by the examples above. I have coined the name "Hoops" for these =

algebras.=20

Any advice would be appreciated.

I would be pleased to e-mail my 480kb Mathematica package GroupMLoops.nb =

and a 430kb notebook HoopsDemo.nb that demonstrates it, to anyone =

interested. They can both be read (but not executed) with Mathreader =

(free from Wolfram) if Mathematica4 is not available.

Roger Beresford.

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<P>(1) How can I extract the defining relationships for a group from =

GAP? Using=20

gmmnn as shorthand for SmallGroup(mm,nn), I particularly need them for =

g2003,=20

g2704, fifteen of the 32-element groups, g3603, & g3609. I have =

written a=20

Mathematica package that handles small groups and loops (up to 120 =

elements=20

without difficulty), as Cayley tables. Any isomorph of a group =

with less=20

than 64 elements (and their subgroups) can be identified. Other =

properties,=20

compositions, etc. are handled. Most of these groups can be =

created (as=20

Cayley tables) by this package, (e.g. SL(2,5) from the permutations =

given in the=20

GAP Tutorial, section 6.2) but I have failed to find relationships to =

create=20

those mentioned.</P>

<P>(2) My internet searches have failed to find anything on the =

following=20

subject, so I am taking the liberty of "asking the experts" via GAP.</P>

<P>Is there a literature and established nomenclature for "Signed =

Tables"=20

(?hypogroups, grouplets, groups "up-to-a-sign"?). These have Cayley =

tables with=20

signed products. The multiplication tables for many algebras are signed =

tables -=20

their structure constants are signed single elements. Many such algebras =

create=20

group tables on composition with some groups (allowing for signs); they =

are=20

"collapsed" from groups via equivalence relationships, an (m)x(m) group =

table=20

becoming an (m/n)x(n/n) algebra. I am cataloguing them in an extension =

of the=20

GAP Group Atlas.</P>

<P>Examples of signed table algebras, with my identifiers:-</P>

<P>Quaternion Pauli=20

sigma =20

Davenport =20

Clifford(2)</P>

<P>g0405  =

; =20

g0410 &n=

bsp; =20

g0408 &n=

bsp; =20

g0409</P>

<P>1 2 3 4 =20

1 2 3 =20

4 1 2 =20

3 4 1 =

2 =20

3 4</P>

<P>2 -1 4 -3 =20

2 1 -4i =20

3i 2 =20

-1 4 -3 =

=20

2 1 4 3</P>

<P>3 -4 -1 2 =

3 =20

4i 1 =

-2i =20

3 4 -1 -2 =20

3 -4 1 -2</P>

<P>4 3 -2 -1 =

4 =20

-3i 2i =

1 =20

4 -3 -2 =

1 =20

4 -3 2 -1</P>

<P>Using ":+" for composition allowing for signs, these compose as =

follows:-=20

g0405:+g0201->g0805; g0410:+g0402->g1613; g0408:+g0201->g0802;=20

g0409:+g0402->g0803. I am studying the algebras that employ =

conservative (in=20

Frobenius's sense i.e all groups plus octonions) Moufang loops and =

signed tables=20

as multiplication tables. They have many interesting properties =

(including, pace=20

Dr Neuebuser) renormalisation as constrained sub-algebras. They provide =

a=20

general framework for many algebras, as evinced by the examples above. I =

have=20

coined the name "Hoops" for these algebras. </P>

<P>Any advice would be appreciated.</P>

<P>I would be pleased to e-mail my 480kb Mathematica package =

GroupMLoops.nb and=20

a 430kb notebook HoopsDemo.nb that demonstrates it, to anyone =

interested. They=20

can both be read (but not executed) with Mathreader (free from Wolfram) =

if=20

Mathematica4 is not available.</P>

<P>Roger Beresford.</P></FONT></FONT></DIV></BODY></HTML>

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