COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

214 P. G. Ruud and R. Keown
4. Current computer programs. During the study and analysis of finite
groups at Texas A &M University, several computer programs have been
written. These programs are all written in FORTRAN IV language and are
operational on the IBM 7094 digital computer. The following paragraphs
are devoted to a discussion of the capabilities of available programs the
extent to which each has been used and the relative merits or demerits of
each.
(1) A program has been written to test whether a Cayley table presentation
is in fact a group by appealing to the group axioms. This program is
completely general and the size of the group that can be tested is limited
only by the machine storage capacity. The highest order group tested on
this program to date is a group of order 64. Should the set being tested fail
to satisfy the requirements of a group, the program indicates which axioms
were violated and which elements of the set failed to comply.
(2) Given the Cayley table of a finite group, a program was written to
determine how many conjugacy classes the group has and to provide a listing
of the elements in each class.
(3) A program was written to determine the order of each element of the
group.
(4) A program was written which first determines all subgroups of a given
group when provided with the Cayley table of the group in question. This is
accomplished by using the test program to verify which combinations of
elements are in fact groups. It then determines which subgroups are normal.
Finally, using the normal subgroups, the corresponding factor groups are
computed. The total output from this program for a given group includes
all proper subgroups, indicates which subgroups are normal and gives the
Cayley table of the corresponding factor group.
This program has been run for groups only as high as order 16. It is not
efficient in the sense of computer time for groups of higher order.
(5) As noted in Q 2, a program was written to construct the corresponding
group Cayley table from the generating permutations for the group. This
program has been used to construct the Cayley tables of all groups of order
2”, n < 6, using the permutations provided in the work of Hall and Senior [5].
(6) The procedure discussed in $2 for computing Cayley tables of
2-groups from the generators and commutator relations was programmed.
This program has been used to generate the Cayley tables for all groups of
order 2”, n < 5, and some of the groups of order 26.
(7) In the analysis of a particular group, it is beneficial if the student can
examine different Cayley table presentations of the same group. Some of the
variations studied were rearrangements of the table according to element
order, conjugacy classes or by positioning a particular normal subgroup of
index four or less in the first part of the table. A program was written to
produce the transformed Cayley table from the original by providing the
computer with the desired isomorphism and the original Cayley table.
Irreducible representations of finite groups
(8) An algorithm similar to the one discussed ins 3 for computing irreducible
representations was programmed. This algorithm differed only in that
the programmed version did not permit the calculation of two or fourdimensional
representations which arose from self-conjugate two-dimensional
representations of the subgroup of index two. Consequently, this
program cannot be used for those groups of order sixty-four where the
above situation arises. The program does calculate one element from each
class of equivalent irreducible representations for groups of order 2”,
IZ < 5. The program is contingent only upon having a suitable Cayley table
presentation of the group available. This poses no restriction in 2” since any
group may be appropriately transformed using the program in paragraph
(7) above. The logic of the procedure is to use the known irreducible representations
of the group of order 2l to obtain those of the group of order 22,
then use these representations of 22 just determined to obtain those of the
next subgroup and so on to the group of order 2”. The output from the
program consists of the matrix representations evaluated at each element
of the group. All groups of order 2”, n < 5, have been subjected to this
program with appropriate results obtained. Approximately 3& minutes of
computer time were used. The same program has been used to calculate the
irreducible representations of some of the groups of order 64.
In actual practice, many of these programs are used simultaneously with
results from one conveyed to others as necessary. For example, the 3+
minutes of computer time above included generating the Cayley tables from
the generating permutations, verifying that the result was in fact a group
satisfying the generating relations, and determining the conjugacy class
structure and the irreducible representations.
As this study of finite groups, and in particular 2-groups, continues, several
new problems are being contemplated. Clearly a modification of the
representation program to carry out the complete algorithm of Q 3 is immediate.
The method of 9 2 for generating Cayley tables looks promising in
considering the step towards 2-groups of order 128. Now that irreducible
representations are available for other 2-groups, a study of their subgroup
structure and other properties is simplified. The question of whether the
representation algorithm would apply with reasonable modification to
groups of order p”, p a prime, also merits consideration.
Readers who might be interested in programs or results of the foregoing
are invited to contact the authors at Texas A&M University, College
Station, Texas, or at the University of Arkansas, Fayetteville, Arkansas.
REFERENCES
1. H. BOERNER: Representations of Groups (John Wiley & Sons, New York, 1963).
2. C. BROTT: Diplomarbeit, Kiel, 1966, unpublished.
3. C. W. CURTIS and I. REINER: Representation Theory of Finite Groups and Associative
Algebras (John Wiley & Sons, New York, 1963).
CPA 15
215