COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

190 A. L. Tritter
With G = B(4, 8) generated by go, gl, . . . , g7, the commutator
[[[go, g11, k2, g311, &?4 g51, k6, g7111
lies in G(3) modulo commutators of order higher than 8.
It is sufficient to place this one commutator, since both GC31 and Gs are
plainly Se-modules (S’s acting by permuting the g,), and commutators of the
same apparent form, but in which not all generators which appear are distinct,
can be taken to be homomorphic images of commutators of this form
with all generators distinct.
Professor G. Higman, to whom I am indebted for this problem, has
shown [2] that this group-theoretic question is equivalent to the following
module-theoretic question (which arises upon examination of the associated
Lie ring of G):
In the free Lie ring of characteristic 2 on 8 generators x0, xl, x2, . . ., x7
we consider the element c x0 xl0 x20 . . . ~7~ (multiplications to be performed
from left to right;, where the 0 are precisely those permutations
(there are 1312 such) on the integers 1,2, . . . , 7 for which (i+ l)o< io for
no more than two values of i.
Letting S’s act by permuting the integers 0, 1, 2, . . . , 7 occurring in the
subscripts on the xi, we generate an &module from this one element,
and we then close this &-module under addition (in the ring), yielding
an additive S,-module.
Question-does the element (((xO~~)(x2~3))((x~xg)(xg~7))) lie in this
additive &-module ?
This paper is concerned with the use of the I.C. T. Atlas, located at Chilton,
Berkshire, and sponsored by the Atlas Computer Laboratory of the U.K.
Science Research Council, to answer this question.
2. General approach. To search for an element in a finite additive module
(and this one has no more than 2s1 elements) is a task most straightforwardly
accomplished by representing the module as a finite-dimensional
vector space, obtaining a basis, and seeing whether the “target” element is
linearly dependent upon this basis. It is clear that all the ring elements which
interest us, whether target or “data”, are balanced homogeneous elements
of weight 8; it is therefore true that the space they span lies within that
generated by all the balanced homogeneous elements of weight 8, equivalently
by the left normed monomials in which the xi appear once each. The
5040 balanced left-normed Lie monomials of weight 8 with x0 appearing
first are known on Lie ring-theoretic grounds to be linearly independent
and to generate (additively) all these monomials, and hence any element of
that part of the ring on which our attention is focused has a unique expressionas
a sum of terms drawn from among them; this expression may be
A module-theoretic computation 191
found by repeated application of the second and third of the “Jacobi identities”
appearing in the customary definition of a Lie ring :
(i) aa = 0 for all elements a
(ii) abfba = 0 for all elements a and b
(iii) a(bc) + b(ca) + c(ab) = 0 for all elements a, b and c.
Thus, our problem splits into two parts in a completely natural way. We
must, first, represent the data by a matrix, 8! by 7!, over the field with 2 elements
(each row giving the expression of a single data element in terms of
the balanced left-normed Lie monomials of weight 8 with x0 at the far left),
and, second, triangularize this matrix, meanwhile searching for proof that
the target vector is, or is not, representable as a sum of rows of the matrix.
We observe that, as representations are unique, no question can arise as
to whether we shall recognize the target when we see it, and that the one ring
element from which all other data elements are generated actually arises in
the form desired, namely as a sum of (13 12) balanced left-normed Lie monomials
of weight 8, with ~0 appearing first in each term. Henceforward, we
reserve the word “term” for this rather special sort of term, a balanced leftnormed
Lie monomial of weight 8 on the letters x0, x1, x2, . . ., x7, with
x0 appearing at the far left.
3. Generating the matrix. The module appearing in the question we are
dealing with has been embedded in the vector space of dimension 7! over
G&‘(2) and is therefore of dimension no more than 7!, and the addition of
the vector space is the ring-addition. But the second operation (upon terms)
with which we are concerned is not the second ring operation, Lie multiplication,
but is rather that operation upon terms induced by permuting the
ring generators x0, x1, x2, . . . , x7. Now it is self-evident that any permutation
of the generators which fixes x0 merely induces a permutation of terms,
but a hand-calculation upon a few examples of the effect of a permutation
of generators not fixing x0 will easily convince the reader that the situation
here is not so simple. We therefore choose an 8-cycle R from S, and represent
every element of S3 as the composition of some power of R with a permutation
fixing 0 (remember, we think of Ss as acting on the subscripts, not
the generators).
The single generator of the &-module we are producing leads to a total
(including itself) of 8 elements if we construct the R-module it generates,
and these 8 elements generate as &-module the structure we want; here 5’7
is the subgroup of S3 which fixes 0. But the action of ST upon terms is
merely to permute them in the obvious way, and it is therefore the case that,
once we have 8 generators for an &-module instead of only one for an
&-module, the two operations we need are simply the addition and multiplication
of the group ring of S7 over GF(2). It is furthermore true that the
full significance for our structure of the Jacobi identities will have been