COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

200 John J. Cannon
stack. The Cayley table may then be stored compactly as a Latin square
with several entries stored in a single machine word. In this manner the
multiplication tables of groups of order up to 380 may be kept in-the core
of a 32K machine. Only when group elements are being output is reference
made to the stack of group words.
The user has the option of outputting either the generators and relations
of each subgroup or the elements of the subgroup or both.
So far the program has been run successfully for a number of small groups
(all of order less than 100). Before larger groups can be run more sets of
generators and relations have to be checked and included in the program.
Investigation of groups of ordeq#,p =- 2. R. James has been enumerating
the p-groups of order p6, p > 2, checking the work of Easterfield, by the
method of isoclinism.
Isoclinism [2, 3] splits the possible sets of generators and relations into
classes so that all members of a class have the same commutator relations.
The problem then is to find all the non-isomorphic groups within each
isoclinism class.
We may suppose that isomorphism is an automorphism and apply a
general automorphism to a general set of generators and relations (remembering
that commutator relations are invariant within an isoclinism class).
This will give rise to a set of relations on the integers mod n, for some n
(obtained by equating indices of each generator before and after the
automorphism). For p-groups, n = pr for some r. These relations can be
expressed as an equivalence relation on a set of matrices over GF(p).
The equivalence classes of these matrices give the non-isomorphic groups
of this isoclinism class.
Difficulties arise in the determination of a set of equivalence class representatives
for general p and it is to this problem that a computer has been
applied. Using a computer it is simple to calculate the equivalence classes
and to select a suitable equivalence class representative (an element which
will give the corresponding generators and relations in the simplest possible
form) for each class, for the first few primes. It is then usually easy to
write down a set of equivalence class representatives for general p.
The groups of order p5 (p > 2) were checked by this method and the
groups of orderpe (p * 2) are being found.
Hughes’ conjecture and commutator calculations. Consider a group G of
order p” where p is a prime. Take the subgroup H of G generated by the
elements of G having order greater than p. We suppose that G has some
elements whose orders are greater than p. Then Hughes’ conjecture is that
H is of index 1 or p in C. It is true for p = 2, 3.
G. E. Wall has shown that the conjecture is false for p if a certain expression
E is zero in a Lie algebra generated by two elements of nilpotency
class 2p- 1, satisfying the (p - 1)th Engel condition, over GF(p).
Combinatorial and symbol manipulation programs 201
Hand calculations carried out by Wall show that the conjecture is false
for p = 5. As the calculations are extremely long and tedious, it was
decided to develop programs to verify the calculations for p = 5, and to
carry them out for p = 7 and 11.
The problem will be considered in four parts:
(1) Determination of a basis for the two-generator free Lie algebra of
nilpotency class 2p - 1.
(2) Determination of a basis of the algebra in (1) with the (p- 1)th
Engel condition imposed.
(3) Calculation of the expression E.
(4) Expression of E in terms of the basis elements found in (2).
These will now be considered in turn.
(1) Let 5 and 7 denote the two generators of the algebra. We define the
weight of an arbitrary element of the algebra as follows: The elements 5
and 7 are of weight one. If A = (P, Q) is an element of the algebra, then
weight A = weight P+weight Q.
Basic commutators are next defined together with an ordering (0 on
them. The elements 5 and 7 are basic. Under the ordering all basic commutators
of weight w come after those of weight w- 1. Ordering is arbitrary
among basic commutators of weight w, but once an ordering is chosen it
must be adhered to. A commutator C = (A, B), A = (P, Q), where A
and B are basic, is basic if A =- B and B 3 Q.
The elements of weight w form a subspace of the algebra and a basis
of this subspace is provided by the basic commutators of weight w [4].
Given an element of the algebra, (A, B), where A = Z&C’,, B = ZpjC”,
Ci, C’ basic, we describe a collection process [5].
’
(i) Put (A, B) = zlipj (Ci, Cj)*
(ii) If Ci and C’ are basic, put
ta) (Ci9 cj) = O if cg.Z Cip tb) tci9 cj) = - (Cj, Ci) if Ci ( Cj,
(C) (Ci, Cj) = (Ci, Cj) if Ci > C* J’
(iii) If Ci * Cj are basic and Ci = (Pi, Qi), put
(a) (Ci, C+‘> = ((pi, Qi>, C’I
if Cj * Qi,
Cb) Cci9ccj> = -((Qi, cj), pi) +((pi, cj), Qi> if Cj < Qi*
(iv) Return to (i) and repeat the process until (A, B) is expressed as a
linear combination of basic commutators.
A program has been developed which calculates the basic commutators