COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

202 John J. Cannon
of weight w by combining all the basic commutators of weight w- 1 with
5 and p in turn, and applying the collection process.
The basic commutators are placed in a stack as they are formed and so
the order of occurrence of the commutators in this stack gives a suitable
ordering of the basic commutators. Each basic commutator (P, Q) is stored
in the stack as a number pair (p, q), where p is the integer giving the
position of the basic commutator p in the stack and similarly for q.
For the collection process linear combinations of commutators are
stored in a list structure. List elements consist of two consecutive words the
first of which contains the coefficient of the present term and a pointer
to the next term, while the second contains the commutator in the form
((p, q), r). New list elements may be obtained from a free space list.
When the Jacobi identity is applied to commutators of weight w, one
requires A = (P, Q), P, Q basic, weight A less than w, to be expressed in
terms of basic commutators. So to avoid much recalculation, after all
the basic commutators of weight w have been found, all products of
pairs of basic commutators giving linear combinations of basic commutators
of weight w are calculated and entered in a table. It is possible to arrange
the table so that no space is wasted and so that products can be looked up
quickly. In the table commutators are again represented in a list structure,
this time, however, using one word list elements. Each list element contains
the coefficient of the present term, the number giving the position of the
commutator in the stack of basic commutators, and a pointer giving the
address of the next term.
On a KDF9 with a 16K store, the 2538 basic commutators of weights
equal to or less than 14 were found in 5 minutes. When the store is increased
to 32K, the machine will be able to find the 4720 basic commutators
of weights equal to or less than 15. I hope also to express all terms of the
Campbell-Baker-Hausdorff formula, of weights equal to or less than 15,
in terms of basic commutators.
(2) The (p- 1)th Engel condition states that
(---(A,f)--_-,p-0
P-l
where B is an arbitrary element of the Lie algebra.
The basic commutators of weights less than p remain independent.
For weights greater than p- 1 one first finds the basic commutators as in
(I), and then derives all the relations between them generated by the Engel
condition. These relations give rise to a homogeneous system of linear
equations in the basic commutators over GF(p), which upon solution
gives a linearly independent set of basis elements, i.e. Engel basic commutators.
The programming is tedious but straightforward.
Combinatorial and symbol manipulation programs 203
(3) The expression E is constructed as follows (all variables are noncommutative
and the polynomials are over GF(j) from (iv) on):
(i) a(X, Y) = ((---(Y,X),X) --- )X)where(A, B)=AB-B-4.
-_1cp-
1 X’s
(ii) b(X, Y) = coefficient of 1p-l p in (1XfpY)P.
(iii) c(X, Y) = b(X, Y)- a(X, Y).
(iv> 4X Y> = :zI&
(v) e(X, Y) = 1:: (X+ qi d(x, Y) (X+ V’-i--2-
0) (X, Y) where (i) denotes a certain ith
derivative.
(vi) f(X, Y) = set of terms of e(X, Y) involving (p- 1) X’s and (p- 1) Y’s.
Now put X = E, Y = /Z where ab = (a, b). (The Lie algebra product.)
(vii) E = zj-( ;, ,c>.
A powerful programming system was developed with the ability to
handle non-commutative as well as commutative polynomials. It is hoped
to publish a description of this system shortly.
As an illustration, the polynomial f(X, Y) for p = 7 contains about
840 terms and took 3 minutes machine time for its construction.
(4) Straightforward.
The hand calculations for p = 5 were verified and a new counterexample
found for p = 7. It is hoped to run p = 11.
Acknowledgments. I would like to thank Dr. Neubiiser for valuable
suggestions concerning the subgroup lattice programs and Professor Wall
who provided the initial motivation for the development of the programs
associated with Hughes’ conjecture.
REFERENCES
1. J. NEUB~~SER : Untersuchungen des Untergruppenverbandes endlicher Gruppen
auf einer programmgesteuerten Dualmaschine. Num. Math. 2 (1960), 280-292.
2. P. HALL: Classification of prime power groups. J. reine angew. Math. 182 (1940),
130-141.
3. M. HALL and J. SENIOR: The Groups of Order 2” (n =S 6) (Macmillan, New York,
1964).
4. M. HALL: The Theory of Groups (Macmillan, New York, 1959).
5. M. HALL: A basis for free Lie rings and higher commutators in free groups. Proc.
Amer. Math. Sot. 1 (1950), 575-581.