COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

148 Marshall Hall Jr.
otherwise no duplications in the lists. If we did not use John Thompson’s
unpublished results (l), apart from lists C and D we would also have to
consider all 62,500 multiples of 16. This would add a large number of
orders to be examined, but probably not very many difficult ones, since
in practice the orders with only high powers of small primes seem to be
the most difficult to eliminate.
From the result (11) of Brauer and Reynolds, if g is divisible by a prime
p Z- 100 then as g -= l,OOO,OOO, p -Z g’” and so G is necessarily PSL&)
or PSL2(2”) where p = 2”+ 1 is a Fermat prime. Hence we may assume
that g is divisible by no prime greater than 97.
At this stage we can divide our search into two parts. For the first part
g is divisible by a prime p in the range 37 == p < 97. For the second part
every prime dividing g is at most 31. The first part is far easier. We have
a prime p dividing g where p4 =- g, since 374 = 1,874,161 =- g. Suppose
first p3jg. Then G has 1 +kp Sylow subgroups S(p) and (1 +kp)p3[ g -= p*,
whence k = 0 and so S(p) a G and G is not simple. Next suppose that p2jg,
and that G has 1 + kp S(p)‘s. Then (1 + kp)p21g < p4 and so 1 + kp c p2.
Here an S(p) of order p2 is necessarily Abelian. As 1 -t- kp -G p2, two S(p)‘s
have an intersection of order p. For if PO n PI = 1, where PO and PI are
two distinct S(p)‘s, then PI would have p2 distinct conjugates under PO and
G would have at least 1 +p2 =- 1 + kp Sylow subgroups S(P), a conflict.
But then by Brodkey’s result (5) all S(p)‘s intersect in a subgroup of order
p which is normal in G, and so G is not simple. It follows therefore from
our assumption that if g is divisible by a prime p such that p4 =- g, then
only the first power of p divides g.
For the primes p with 37 =Z p < 97 we rely on the Brauer-Reynolds
results of (11) and (12). The number 1 + rp of admissible S(p)‘s was calculated
by (3.1 l), and degrees satisfying (3.2) were found. In g = (1 +rp)pqw
the values of g and w were determined so that g would be divisible by 168
or 48, following Gorenstein and Walter’s results (2) and (3). The details of
these calculations for p = 59 are given in Example 1 of 9 5.
The Stanton condition of (8) and the principle of block separation are
applicable. An illustration of block separation is given in Example 2 of 0 5.
All orders multiples of primes p, 37