COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

222 N. S. Mendelsohn
(b) The cube of a commutator can be expressed as a product of two
commutators, e.g.
(a, by = (a-lb-la-lba, a-lb-%z)(ubu-lb-Q& b).
(c) Each of the products (a, b) (c, a) and (a, b) (b, c) can be expressed as
a single commutator.
(d) c-~(cx)~x-~ = (xc*, ~-lx-~).
This last relation leads to a very interesting theorem of group theory.
THEOREM. Let G be a group for which every commutator has order 1 or 3.
Then G’ is periodic.
Remark. This theorem was proved by a colleague of mine, N. D. Gupta.
He used a more complicated lemma than that used in the proof below.
Proof. From the identity c-~(cx)~x-~ = (xc2, ~-lx-~) it follows that
(cx)~ = c*x3 where c and c* are commutators. Hence if cl, c2, cs, . . ., c,
are commutators (crc~ . . . c,)~ = c~*cz*. . . ct-r. By iteration (cite . . .
&J3m = 1.
Concluding remarks. The above examples indicate that there can be a
fruitful symbiosis between machine and mathematician. The author would
also point out that it is well worth while for him to scan the output of a
computer even though it may appear random and disconnected. There is
the possibility of real discovery.
REFERENCES
1. E. N. GILBERT: Latin squares which contain no repeated digrams. SIAM Rev. 7
(1965), 189-198.
2. B. GORDON: Sequences in groups with distinct partial products. Pacific J. Math. 2
(1961), 1309-1313.
3. R. G. SANDLER: The collineation groups of free planes II: A presentation for the
group G,. Proc. Amer. Math. 5’0~. 16 (1965), 181-186.
Construction and analysis of non-equivalent
finite semigroups
ROBERT J. PLEMMONS
1. Introduction. In searching for examples of finite algebraic systems
that satisfy certain identities or have specific properties, it is often convenient
to have available a listing of all non-equivalent (i.e., non-isomorphic
or anti-isomorphic) systems of given types and orders, together with
information concerning their properties. This paper is concerned with the
development of algorithms used to design computer programs for the
purpose of constructing and analyzing certain systems such as groupoids
and semigroups, having small orders. Of course, the basic problem in
such projects is to develop an efficient algorithm to construct one representative
system from each class of those that are either isomorphic or
anti-isomorphic.
Digital computers were first applied to the construction of non-equivalent
finite semigroups by G. E. Forsythe in 1954 [3], when he constructed all
semigroups of order 4 by use of the computer SWAC, at Los Angeles.
Hand computations [10] had previously yielded those of order N g 3. In
1955, T. S. Motzkin and J. L. Selfridge obtained all semigroups of order 5,
also by using SWAC, and about the same time similar results were obtained
in Japan by hand computations [11]. It was not until 1966 that the results
for N = 6 were obtained at Auburn University [8].
In $2 we develop an algorithm to construct all non-equivalent semigroups
of order N e 6, the results for N = 6 being new. The analysis of
these semigroups is discussed in 0 3. In addition, some applications to
the development of certain theorems about finite semigroups are mentioned,
along with the formulation of an associated conjecture. All notation
and definitions follow [1] and [2].
2. The construction algorithm. As we have mentioned, the problem of
constructing all non-equivalent finite algebraic systems of given type and
order is essentially the problem of efficiently choosing a representative
system from each class of those that are either isomorphic or anti-isomorphic.
It is trivial to construct an algorithm to do this. One needs only
to compute all possible systems of that type, to determine which are isomorphic
or anti-isomorphic and then to choose one system from each
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