COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

A module-theoretic computation related to the
Burnside problem
A. L. TRITTER
ALTHOUGH the Burnside conjecture is known to be true for exponent 4
(i.e. although it is known that finitely generated groups of exponent 4 are
finite), we cannot usefully say that the Burnside problem is settled, even in
this case. For instance, a sharp bound on the order of B(4, n) would be
valuable, and there are perhaps other questions that arise in consideration
of the Burnside problem whose answers would be of interest.
1. Introductory considerations. Defining the lower central series {Gi}
and the derived series {Go} of a group G in the usual way :
GI = G, G+I = [G, Gl,
GC’) = G, G(i+U = [GOI, (JO],
where [H, K] (H& G, KE G, G a group) is the least subgroup of G containing
all commutators h-li?-%k (& H, kE K), we know that, for every
group G and for every natural number n, G(“)E Gsn, But we can deduce
from a result of C. R. B. Wright [1] that, when G is of exponent 4, any
inclusion G(‘)S G, not predicted from this elementary result (i.e. with
2’(s) must lead to a bound for the derived length of G. If we choose
G = B(49 8) and II = 3, what we are saying is this:
(i) we know Gc3) 5 Gs to be true *
(ii) if we could show Gc3)ZGs we could bound the derived length of
every group of exponent 4.
The bound would be applicable to all groups of exponent 4 because of the
natural homomorphisms to groups of exponent 4 on fewer than 8 generators
and the fact that, for any group G, GC31 is generated modulo Gs by commutators
of the form
[[[a03 al,], [&t, a311, [[a43 a519 [a62 a7111,
where the a, are among the generators of G, so that no more than 8 distinct
generators of G could be present in any one of these commutators. We wish,
therefore, to show that:
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