COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
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COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA. COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

thales.doa.fmph.uniba.sk
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05.01.2013 Views

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

192 A. L. Tritter expressed in the technique we use to apply the operation on terms induced by R, in getting from one generator for an &-module to 8 generators for an ST-module. To sum up, for reasons of computational simplicity we shall not look directly at the Ss-module generated by c ~~~~~~~~ . . . x7o, but we shall see it by looking at the S,-module genera&d by {C XORtxlaRix~Ri. (I . .x7oRI ( 0 4 i -= B}. 4. The R-module. There is a natural mapping from the group ring of Ss over GR(2) onto that part of our Lie ring we have already termed interesting, the balanced homogeneous elements of weight 8, induced by mapping 0 1 2...7 ( a0 al a2 . . . u7 ) onto the balanced left-normed monomial x~~x~~x~~ . . . x,,, clearly of weight 8, and extending linearly. Also, there is a one-to-one mapping onto the group ring of S7 (the subgroup of S’s fixing 0) over GR(2) from this same part of the Lie ring induced in exactly the same way, but requiring the element of the Lie ring to have been expressed in its unique form as a sum of left-normed monomials in which x0 appears first. The composition of these two mappings is a mapping p from the group ring of Ss over GF(2) onto the group ring of ST over GF(2), fully expressing the effect of the Jacobi identities in this part of the Lie ring. It should be observed that the restriction of ,u to the group ring of S, (the subgroup of S’s fixing 0) over GF(2) is the identity mapping. It should now be clear that to construct the mapping on terms induced by R, we need only know how to multiply each element seen as an element @.= ( 1 2 .D. 7\ of s 59 ccl cI.2 ~. , GC7 I 0 1 2 . ..7 0 a1 M2 . . . cc7 of the group ring of Ss over GF(2), by R, and obtain the image under ,U in the result. This, in fact, is precisely what we do. The R we employ (it could have been any B-cycle) is if! O=si

192 A. L. Tritter<br />

expressed in the technique we use to apply the operation on terms induced<br />

by R, in getting from one generator for an &-module to 8 generators for an<br />

ST-module.<br />

To sum up, for reasons of computational simplicity we shall not look<br />

directly at the Ss-module generated by c ~~~~~~~~ . . . x7o, but we shall<br />

see it by looking at the S,-module genera&d by<br />

{C XORtxlaRix~Ri.<br />

(I<br />

. .x7oRI ( 0 4 i -= B}.<br />

4. The R-module. There is a natural mapping from the group ring<br />

of Ss over GR(2) onto that part of our Lie ring we have already termed<br />

interesting, the balanced homogeneous elements of weight 8, induced by<br />

mapping<br />

0 1 2...7<br />

( a0 al a2 . . . u7 )<br />

onto the balanced left-normed monomial x~~x~~x~~ . . . x,,, clearly of<br />

weight 8, and extending linearly. Also, there is a one-to-one mapping onto<br />

the group ring of S7 (the subgroup of S’s fixing 0) over GR(2) from this same<br />

part of the Lie ring induced in exactly the same way, but requiring the element<br />

of the Lie ring to have been expressed in its unique form as a sum of<br />

left-normed monomials in which x0 appears first. The composition of these<br />

two mappings is a mapping p from the group ring of Ss over GF(2) onto<br />

the group ring of ST over GF(2), fully expressing the effect of the Jacobi<br />

identities in this part of the Lie ring. It should be observed that the restriction<br />

of ,u to the group ring of S, (the subgroup of S’s fixing 0) over GF(2)<br />

is the identity mapping.<br />

It should now be clear that to construct the mapping on terms induced by<br />

R, we need only know how to multiply each element<br />

seen as an element<br />

@.=<br />

( 1 2 .D. 7\ of s<br />

59<br />

ccl cI.2 ~. , GC7 I<br />

0 1 2 . ..7<br />

0 a1 M2 . . . cc7<br />

of the group ring of Ss over GF(2), by R, and obtain the image under ,U in<br />

the result. This, in fact, is precisely what we do. The R we employ (it could<br />

have been any B-cycle) is<br />

if! O=si

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