COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

206 P. G. Ruud and R. Keown Irreducible representations of finite groups 207
in which each subgroup Hi is a subgroup of index two in Hi+l, 1 e i =S
r- 1, where G is a 2-group of order 2’. Our irreduci’ole representations are
monomial representations constructed by induction on the normal series
(1.1). Originally the authors believed that each self-conjugate representation
of an Hi in this series would generate the two associated representations
of Hi+1 under multiplication by the appropriate scalar matrix.
This idea proved false. Presently, it is conjectured (not proved) that a
minor modification of the method and program will avoid this error.
The second section of the paper discusses the manner of obtaining the
Cayley table of a 2-group from the defining relations of Hall and Senior.
The third section explains the inductive construction of the irreducible
representations from the Cayley table. The fourth section describes the
programs used in the calculation. Until the early part of 1967, the present
authors were unaware of the very significant work on representation theory
in progress abroad, see Brott [2], Gerhards and Lindenberg [4], Lindenberg
[8], and Neubiiser [10]. Excellent sources of information on the theorems
quoted in this paper are the monographs of Boerner [l], Curtis and
Reiner [3], and Lomont [7].
2. Development of the Cayley tables of 2-groups. The method of calculation
of the irreducible representations of 2-groups described in this paper
depends upon the availability within the computer of the Cayley table of
the group under investigation. The work of Hall and Senior describes each
group of order 2”, 1 =z y1 =z 6, in three ways: “(1) by generators and defining
relations; (2) by generating permutations; and (3) by its lattice of
normal subgroups, together with the identification of every such group
and its factor group”. The generation of the Cayley table from the permutation
presentation of a group appeared to be the most expedient
method, if not the most concise. R. F. Hansen [6] wrote the necessary
program and assumed the burdensome task of preparing the necessary
input for all 2-groups of order greater than 4 and less than 128. All Cayley
tables were computed by this method and checked for accuracy. A very
small number of typographical mistakes in the lists of permutation generators
was apparently uncovered in the process. D. L. Stambaugh [12]
attacked the problem of discovering a satisfactory method of obtaining
the Cayley tables directly from the generators and the defining relations.
An examination of a number of Cayley tables strongly suggests that a
presentation of the generators as regular permutations with degrees equal
to that of the group can be obtained in a direct manner. In order to describe
the method discovered by Stambaugh, it is convenient to briefly discuss
the definition of the groups by means of the Hall-Senior defining relations.
Each group G is described by a set B of generators [bI, . . . , b,] where
the number r is the exponent of 2 in the order 2’ of the group. Every element
g of G has a unique standard expansion,
g = bi’ . . . b:, (2.1)
I
~
I
I
in terms of these generators where each ei, 1 < i 4 r, is either 0 or 1. The
element g with corresponding exponents the set [er, . . . , e,] is numbered
e,2’-l+ . . . +er+l. (2.2)
We adopt the notation Hj for the subgroup (bl,. . . , bj) generated by the
first j generators. The ascending normal series,
{l)cH1 c . . . c H, = G, (2.3)
is the basis of the present calculation of the irreducible representations of G.
The Hall-Senior defining relations are given in terms of a subset [al,. . . , aj]
of B. The squares a!, 1 4 i--j, of these are listed either in terms of central
elements or a’s with smaller subscripts. The collection of all commutators
[ai, a,] = a,yl ai’aia,, 1 g i < m ~j, is similarly described. The
set [al,. . . , aj] is a proper subset of [bl, . . . , b,] when G is not a stem group.
Stem 2-groups are those in which the center Z is contained in the derived
group G’ of G. Additional information is given for non-stem groups enabling
one to extend the information from the set [al,. . . , aj] to the set
h, . . . , b,]. The computational scheme of Stambaugh is based on a set of
defining relations which gives all squares b;, 1~ i < r, and all commutators
[bi, b,], 1 =z i-=m =z r, in terms of generators with smaller subscripts.
Stambaugh writes the commutators in the form
bib, = b,,bixy 1 < i-z m < r, (2.4)
where x is obtained at an earlier stage in an inductive computational scheme.
We refer to the description of each group in terms of the b’s as the
Hall-Senior defining relations although, strictly speaking, these correspond
exactly to those of their monograph only for stem groups.
,
I The set [vr,. . ., v,] of regular permutations corresponding to the set
[h, . . . , b,] of generators of the group G has a number of properties conveniently
discussed together. Each permutation yi is associated with three
sets, Bi, El, and Si, which are defined as follows: Bi consists of the set
[l,. . . ,2’-l] of the first 2’-l integers; Ei consists of the set [2’-l+ 1, . . . ,2’]
of the next 2’-l integers; while 5’i, the union of Bi and Ei, consists of the
first 2’ integers. Each yi is determined by a permutation pi defined on the
set Si where it maps Bi in a one-one fashion on Ei and conversely. Consequently,
its inverse p;l has the same domain, but the opposite effects
on Bi and Ei. Each permutation pi is extended to the set of the first 2
integers by means of a family of permutations [cl,. . . , crml]. The permutation
ci is defined on the set Si+l in the following fashion:
S(k) = k+2’, k E Bi+l,
ci(k) = k-2’, k E Ei+ 1,
l=zi=r-1. (2.5)
The ith permutation pi, defined on the set Si, is extended by successive