Date: Fri, 25 May 2007 03:40:22 -0600
From: Michael Kinyon <mkinyon@math.du.edu>
To: Bob Veroff <veroff@cs.unm.edu>
Cc: William McCune <mccune@cs.unm.edu>
Subject: [SAX] summary

As you requested, here is a summary of what I was able to do
with the single axiom candidates. This is in reference to the
list of 36 leftovers at

http://www.cs.unm.edu/~mccune/projects/gtsax/fg/dependencies

The candidates I was able to eliminate are 1, 2, 5, 19, 22, 23.

My approach was to try imposing conditions which hold in groups,
and see for which candidates those conditions would be enough
to force it to be a group. If the conditions still didn't tip
the balance, then I would hunt for counterexamples satisfying
the extra conditions.

Firstly, I imposed the condition f(x,g(x)) = 0, and then noticed
this in the output:  g(g(g(g(g(g(x)))))) = x. That suggested a
model of order divisible by 6. Mace4 found a large model right away,
and then after I discussed it with Bill, I went back and found a model
of order 6:

interpretation( 6, [number=1, seconds=0], [
       function(g(_), [ 0, 2, 3, 1, 5, 4 ]),
       function(f(_,_), [
              0, 3, 1, 2, 4, 5,
              1, 4, 0, 5, 3, 2,
              2, 5, 4, 0, 1, 3,
              3, 0, 5, 4, 2, 1,
              4, 1, 2, 3, 5, 0,
              5, 2, 3, 1, 0, 4 ])
]).

By the exact same approach, I found another model which eliminates
candidates 19, 22, and 23.

interpretation( 6, [number=1, seconds=0], [
       function(g(_), [ 0, 3, 1, 2, 5, 4 ]),
       function(f(_,_), [
              0, 1, 2, 3, 4, 5,
              2, 4, 0, 5, 1, 3,
              3, 5, 4, 0, 2, 1,
              1, 0, 5, 4, 3, 2,
              4, 2, 3, 1, 5, 0,
              5, 3, 1, 2, 0, 4 ])
]).


So that takes the list down to 30 candidates. Like I said, it's a sliver.

That was as far as I got with eliminating models. Jumping a bit out
of chronological order, I found some more dependencies between models.
In particular, candidates 10, 20, 29, 33 are equivalent. So there are
really 27 candidates left (including the ones which complete, but
for which no finite model is known). Bill then checked and found no
additional dependencies other than those listed on the web page.

I then tried to see for which candidates f must be a quasigroup operation
and g must be a permutation. I found that these conditions hold in 
33/10/20/29,
in 3, in 9, and in 11. In the others, searches fail or I eventually killed
the job.

Using some standard techniques of quasigroup theory, I characterized
the 33 gang. Here is what I found:

Suppose (G,f,g) satisfies 33.
Then there exists a group structure (G,+,',0) (not necessarily commutative),
and  two automorphisms a and b of the group such that

f(x,y) = a(x) + b(y) and g(x) = a^{-1} b^{-1} a b^{-1}(x').

The automorphisms must satisfy some conditions:

1.    a2 b2 = b2 a
2.    a^{-1} b a(x) + b a^{-1} b(x') + a2(x) + x' = 0  for all x
3.    a(x) + b a^{-1} b^{-1} a b(x') is in the center of G for all x

Conversely, suppose (G,+,',0) is a group and suppose a and b are 
automorphisms
satisfying 1, 2, 3. Define f and g as above. Then 33 holds.

I still don't have such a quasigroup, and I suspect that a model would be
quite large.

I wasn't able to get as much information out of 3, 9, or 11.

That's about as far as I got. I stopped working on it when I ran out of 
ideas
and got busy with other stuff.

My gut feeling is that none of the remaining candidates are single 
axioms for group theory.

Michael