COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

300 Lowell J. Paige Non-associative algebras 301
groups in 1951 [2] and Professor M. Hall generalized the results for the
abelian case in 1952 [3].
It was at this time (1953) that I sought help from the computer and obtained
a detailed analysis of complete mappings for groups of small order.
The memory capacity of SWAC at that time prevented any large-scale
analysis, but the results were of such a nature that Professor Hall and I
reconsidered the problem. We gave a complete solution to the problem
for solvable groups and stated the following conjecture in 1955 [4]:
CONJECTURE: A jinite group G whose Sylow 2-subgroup is non-cyclic possesses
a complete mapping.
This conjecture has never been verified nor has a counterexample been
found. Computers could certainly provide more evidence, but the solution
of the problem has rather dubious applications to the original problem of
projective planes. There is, however, an interesting footnote to the conjecture.
I felt that I could provide a solution if the following published problem
of 1954 were true:
PROBLEM. Let G be a finite group and Sz a Sylow 2-subgroup. In the coset
decomposition of G by Sz, does there exist an element of odd order in each
coset ?
The problem remained unsolved until Professor John Thompson provided
a counterexample in 1965. His example was the group of 2 x 2 unimodular
matrices over the Galois Field GF(53). It is easy to see that the Sylow
2-subgroup of Thompson’s example is non-cyclic, and there is reason to
suppose that this example might provide a counter-example to our original
conjecture; however, the order of this group (148,824) makes it seem unlikely
that even today’s computers would be capable of providing the answer.
My experience with computers and 10 x 10 orthogonal lattice squares
was not a particularly successful venture. In 1958, I wrote, “consequently,
the total time necessary to do an exhaustive search for latin squares orthogonal
to our example would be approximately 4.8 X 1On machine hours”.
Perhaps the time computation was correct but we are all well aware that the
counter-example to Euler’s conjecture was provided the next year.
An example of a computer-provided counter-example to another of Euler’s
conjectures occurred last January when L. J. Lander and T. R. Parkin
published the following numerical relation [5] :
275+-845+1105+133~ = 1445.
Let me turn now to an example from loop theory for a more favorable
experience with a “machine suggested” conjecture. First, a brief review of
the pertinent loop theory.
Two loops (G, .) and (H, X ) are said to be isotopic if there exists a triple
of bijections (cc, j, y) of G to H such that
WX(YB) = (X-Y&
for all x, y of G . Professor R. H. Bruck raised the following problem in
1958 [6]:
Find necessary and sufficient conditions upon a loop G in order that every
loop isotopic to G is isomorphic to G.
The answer to this problem, as was pointed out by Bruck, would have
interesting interpretations to projective planes.
It is well known that a group G has the property that it is isomorphic to
all of its loop isotopes. A student at the University of Wisconsin, working
under the direction of Professor H. Schneider, examined all loops of order
7 on a computer and discovered that the only loop isomorphic to all of its
loop isotopes was the cyclic group. He has proved subsequently that any
loop of prime order satisfying the property that it was isomorphic to all of
its loop isotopes must be the cyclic group. Moreover, I understand that he is
now considering a generalization of this result to loops of prime power
order.
Another example which comes to mind involves a question raised by Professor
T. A. Springer [7] concerning elements in a Chevalley group; specifically,
“Is the centralizer, G,, of a regular unipotent element .Y, an abelian
group ?”
Dr. B. Lou, working under the direction of Professor Steinberg at the
University of California, Los Angeles, has given the answer to this question
in those cases left open by Springer, and a considerable portion of her
work in the associated Lie algebras was done on a computer.
It would be a matter of serious negligence in surveying the applications of
computers to non-associative systems if one were not to mention the work
of Professor Kleinfeld [8] on Veblen-Wedderburn systems with 16 elements,
or that of Professor R. Walker [9] in extending these results to a listing of
finite division algebras with 32 elements. Finally, the work of Professor
D. Knuth [10] in providing new finite division algebras is an excellent
example of computer assisted research.
3. Jordan algebra identities. The problem of determining identities
satisfied by the elements of a non-associative algebra is one area in which
the computer could be expected to make research contributions. For example,
Professor R. Brown of the University of California has discovered an
identity in one variable for certain algebras arising in his investigation of
possible representations of the Lie group associated with ET.
Professor M. Osborne of the University of Wisconsin has published all of
the possible identities of degree 4 or less for commutative algebras. These
were done without the aid of a computer. However, Professor Koecker of
Munich has sought computer assistance in determining all possible identities
of a non-associative algebra where the degree of the identity is restricted
to degree 6 or less. The early estimates of the machine time required for