COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

186 E. Krause and K. Weston
mk}, mi = W$(Xl, . . ., x,), we show here an easy method to calculate A,
from the matrices M6(g(xl)), . . . , M,(g(x,J).
IfIi, . . . . & is the natural basis for A, then by (1)
Ai @ Aj = M&(mj)) X Ai
(i.e. 1i@ & = the ith column of M&(mj))). Thus a multiplication table
for A, is easily calculated by observing the columns of the matrices
~,tdml)), . . -, M&(mk)).
Because of our convention of writing operators on the left, the inner derivation
g is a Lie anti-homomorphism of L into the algebra of linear operators
:
de-m) = [g(m), g(e)1 = - Me>, g(m>l, e, m EL, where V’, Sl = TS’-- ST,
S, T linear operators. Also MP(g(e.m)) = [MB(g(m)), h$(g(e))] or more
generally
%&tel, . , . , ej)> = t- l)j+1W&ted)9 ~&te2)), . . -, ~&dej))l. (2)
Consequently (2) affords a simple formula for calculating M&(mi)),
i= 1,. . . , k from the matrices MB(g(xr)), . . . , J@g(x,)).
Therefore, with restrictions on the size of k of course, one can use a computer
to calculate and even print the multiplication table of A, given the
matrices ~&d~l)), . . ., ~&tx,)).
Algorithm for L(p, n, m) (m -= p). Designate L(p, n, m) by L; since
m-=p, L(p, n, m) is nilpotent of class c [2]. We wish to use Theorems 1 and 2
to determine a basis p for L/Lk (k = 1,2, . . . , c) from the bases of L/L=, . . . ,
Lkm2JLk-l. Therefore A, 2 L[Lk by Theorem 1.
Suppose L is generated by x1, . . . , x, and LIx/La+l (a = 1, . . . , x- 1)
has a basis consisting of monomials p@ = {m,, l+La+l, . . . , m,,jl+La+l}
m,, ji = m, jc (Xi, . . . , x,J. Next select any set of monomials
Pk-i = {mk-1, l+Lk, . . ., mk-l,jkel+Lk}, mk-l,jr = mk-1, ji(Xi, . . . , Xn)
which span L k - ’ IL k . For example &-l could COnSist of all of the
monomials with k- 1 factors. Then L/r;” is spanned by
B = {ml,l+Lk, . . . , ml, j,-tLk, . . . . , mk-l,lfLk, . . ., mk-l,jkel+Lk}
and p constitutes a basis if and only if fl&., is a basis. Thus if A, is a Lie
algebra over GF(p) on n generators satisfying the mth Engel condition,
then A, % L/Lk by Theorem 2. Hence, by Theorem 1, fl is a basis, which in
turn implies that p&1 is a basis. If A, fails to satisfy any of the above conditions,
,$+i must be a set of dependent vectors. Thus we have to select a
proper subset spanning Lk-‘/Lk and repeat the process.
This process of course only affords us a check whether a basis has been
found for Lk-l/Lk and does not actually calculate one except by trial and
error. The use of a computer to calculate A, has already been mentioned.
A computer may be used also to scan the multiplication table of A, and
determine which of the conditions of Theorem 2 are fulfilled by A,. A pro-
_
Restricted Burnside group of prime exponent 187
gram for determining whether A, is a Lie algebra on n generators satisfying
the mth Engel condition is on file in the Computing Science Library of the
University of Notre Dame.
REFERENCES
1. E. KRAUSE and K. W ESTON: The restricted Burnside group of exponent 5 (in preparation).
2. A. KOSTRIKIN: On locally nilpotent Lie rings satisfying an Engel condition. Doklady
Akademii Nauk SSSR (1958), 1074-7.