COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

208 P. G. Ruud and R. Keown
products and conjugations to obtain
Yi = Pi[CiPiC;‘l [Ci+lPiCiPi(C~+lCi)-‘IX + . *
X [Cr-IfJiCr-2 e a . Cipi(C,-1 . . e Ci)-‘1.
(2.6)
The permutation p1 is associated with the sets Br, El, and S1 corresponding
to [l], [2], and [l, 21 respectively; it is defined to be the transposi-
tion ’ 2 It is easily checked that p1 fulfills the above requirements.
( 21. 1
The permutation y1 corresponding to bl is obtained by means of equation
(2.6).
The construction of the remaining p’s and y’s is carried out by induction.
Suppose that the definition of the set [yl, . . . , yk- r] of the first k- 1
permutations has been completed. We consider the construction of pk.
We are given the following conditions:
(1)
pk(l) = 2k-1+1,
(2) 3’1Pk = Pkylqlt
. . .
(k) Yk-lpk = PkJ’k-lqk-1,
(kf 1) Pkpk = qk,
(2.7)
which are required to hold on Sk where the set of permutations [yl, . . . ,
Yk-1, 41,. * ., qk] are known.
The preceding kf 1 conditions are written in the form employed in
the program of Stambaugh; however, a description of his method appears
somewhat clearer if the commutation relations are rewritten in the form
pk = (y;lpiyi)qi, 1 =Z i G k- 1. CW
One begins the construction with the knowledge of pk(l), always defined
by (2.71). At the beginning of the ith stage, employing (2.7(i+ l)), pk
has been defined on the set Bi of the first 2’-l integers with pk taking values
in Ek. The problem is to extend the domain of definition of pk to the set
Ei of the next 2’-l integers with pk taking values in Ek. The right factor
qi of the product (yi-‘pkyi)qi maps the set Ei onto itself in a l-l manner.
The next right-most factor yi maps Ei onto Bi where pk is already
defined. The mapping pk itself maps Bi into the set Ek which is carried into
itself by the last mapping yyl in the product. It follows that the definition
of pk has now been extended to the set Bi+l. After condition (2.7k)
iS employed, pk iS defined on Bk with pk taking VdUeS in Ek. This implies
that p;l is defined on Ek and takes its values in Bk. The (k+ 1)th
relation of (2.7) can be rewritten in the form
Pk = qkpkl, (2.9)
Irreducible representations of jinite groups 209
where the permutation on the right is well-defined on the set Ek and assumes
values in Bk. This Completes the definition of pk on the Set Sk. The
definition of yk follows from equation (2.6). This method of construction
is effective for all groups of order 32 and for those groups of order 64
for which it has been employed.
A modification of the method is under consideration in which the back
solutions of (2.7) are used to extend the permutations rather than (2.6).
These equations can be rewritten in the form
(1) pk(l) = zk-‘+l,
(2) PlPk = pkrl,
. . .
6'4 Pk--1pk =Pkrk-1,
(kf 1) Pk = rkpil,
(2.10)
denoting pi and its extensions by the same symbol, where the set [PI,. . . ,
pk-1, rl,. . ., r.& i] of permutations is known on the set Sk-r before the kth
stage of construction. One finds from (2.10) permutations representing
the 2k-1- 1 elements of G following b, in the Cayley table. These appear
in the form
(1) PlPk = Pkrl,
(2)
. . .
(12 . . . (k- 1)) P~PZ . . . pk = pkrlr2 . . . yL-1.
Assume, as an induction hypothesis, that all of the known permutations
agree on Sk-, with the presentations of their corresponding elements in
the left regular presentation of G. The values of the left members of (2.11)
are known for the left regular presentation on the number 1, which implies
that pk, the permutation representing bk in the left regular presentation,
is known on the set Sk-,. Using equation (2.9), one determines
the values of the left regular presentation of bk on the remainder of the
set Sk of the first 2k integers. Solving for the 2k-1 permutations preceding
pk in the form
Pl*.. pj = pkrl . . . 'j/Pi1 p.w
permits their evaluation on the Set Ek and, consequently, on the set Sk.
The determination of the left regular presentations of the 2k-1- 1 elements
following bk on the set Sk can then be completed. This finishes
the kth stage of the construction. Since the induction hypothesis is surely
fulfilled for k= 1, it follows that this method determines the left regular
presentation of the group G. It appears that this method will generate the
Cayley table in substantially less time than the first. It has not yet been
programmed.