> < ^ Date: Sat, 15 Mar 2003 10:53:14 -0000
< ^ From: Roger Beresford <rogerberesford@supanet.com >
> ^ Subject: Group defining relations, "Signed Tables".

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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, &amp; 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&nbsp; group =
with less=20
than 64 elements (and their subgroups) can be identified. Other =
properties,=20
compositions, etc. are handled.&nbsp; 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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Pauli=20
sigma&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=20
&nbsp;Davenport&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=20
&nbsp;Clifford(2)</P>
<P>g0405&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;=20
g0410&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=20
&nbsp;g0408&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;=20
g0409</P>
<P>1&nbsp; &nbsp;2 &nbsp; 3&nbsp;&nbsp; 4&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=20
&nbsp;1&nbsp; &nbsp;&nbsp;2&nbsp;&nbsp; &nbsp;&nbsp;3&nbsp;&nbsp;&nbsp;=20
4&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;1&nbsp; &nbsp;2&nbsp;&nbsp;&nbsp;=20
3&nbsp;&nbsp;&nbsp; 4&nbsp;&nbsp;&nbsp; &nbsp; &nbsp;1&nbsp;&nbsp; =
2&nbsp;&nbsp;=20
3&nbsp;&nbsp;&nbsp; 4</P>
<P>2 &nbsp;-1 &nbsp; 4&nbsp;&nbsp;-3&nbsp;&nbsp;&nbsp; &nbsp;=20
&nbsp;2&nbsp;&nbsp; &nbsp;1&nbsp;&nbsp; -4i&nbsp;=20
&nbsp;3i&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;2&nbsp;=20
&nbsp;-1&nbsp;&nbsp;&nbsp;&nbsp;4&nbsp;&nbsp; -3&nbsp;&nbsp; =
&nbsp;&nbsp;&nbsp;=20
2 &nbsp; 1 &nbsp; 4 &nbsp;&nbsp;3</P>
<P>3 &nbsp;-4&nbsp; -1&nbsp;&nbsp;&nbsp;2&nbsp;&nbsp;&nbsp; &nbsp; =
&nbsp;3&nbsp;=20
&nbsp;4i&nbsp;&nbsp;&nbsp;&nbsp; 1 =
&nbsp;-2i&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=20
&nbsp;3&nbsp;&nbsp; &nbsp;4&nbsp;&nbsp; -1&nbsp; &nbsp;-2&nbsp;&nbsp;=20
&nbsp;&nbsp;&nbsp; 3&nbsp; -4&nbsp; &nbsp;1 &nbsp;-2</P>
<P>4&nbsp;&nbsp; 3&nbsp; -2 &nbsp;-1&nbsp;&nbsp;&nbsp; &nbsp;&nbsp; =
4&nbsp;=20
-3i&nbsp;&nbsp; &nbsp;2i&nbsp;&nbsp;&nbsp; =
1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=20
4&nbsp;&nbsp; -3&nbsp;&nbsp; -2&nbsp;&nbsp; &nbsp; =
1&nbsp;&nbsp;&nbsp;&nbsp;=20
&nbsp;&nbsp;4 &nbsp;-3&nbsp;&nbsp;&nbsp;2&nbsp; -1</P>
<P>Using ":+" for composition allowing for signs, these compose as =
follows:-=20
g0405:+g0201-&gt;g0805; g0410:+g0402-&gt;g1613; g0408:+g0201-&gt;g0802;=20
g0409:+g0402-&gt;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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