COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

142 Marshall Hall Jr. Simple groups of order less than one million 143
(9) Burnside has shown that groups of orders pOqb, p and q primes, are
solvable. A proof of this may be found on page 291 of the writer’s book [23].
(10) It was shown by Burnside that if a Sylow p-subgroup 5’(p) of a group
G is in the center of its normalizer N(p), then G has a normal p-complement
([23], p. 203). Here the normal p-complement K is a normal subgroup of G
such that G/K z S(p). An easy consequence of the Burnside result ([23],
p. 204) is that the order of a simple group is divisible by 12 or by the cube of
the smallest prime dividing its order. Further results depending on the
theory of the transfer ([23], ch. 14) assure the existence of normal subgroups
with p-factor groups. A recent theorem of John Thompson’s [32] gives an
elegant condition for the existence of normal p-complements.
(11) Brauer and Reynolds [l0] have shown that a simple group G whose
order g is divisible by a prime p =-q113 is isomorphic to PSL2(p) where p s-3
or to PSLz(2”) where p = 2”+ 1 is a Fermat prime. Thus these groups are
the unique simple groups of their orders.
In a more refined form they show that if p2 {g and we write g =
pqw(1 +rp) as in (3.1), if it should happen that
(p- 1)(1 +rp> = (VP- l)(up+ 1) (3.6)
has only up- 1 = p- 1 and up + 1 = 1 frp as a solution in integers, then
with S the largest normal subgroup of G of order prime top we have one of
(a) G/S Y PSL4p) and r = 1,
(b) G/S zz PSL42”) where p = 2”f 1, r = ~(JI- 3),
(c) G/S is the metacyclic group of order pq.
(3.7)
This argument depends on the fact that the degreesh, i = 1, . . . , q, and f0
in the principal block B&) divide (p- l)( 1 + rp) and if there is no second
representation in (3.6) then A and tfo can only take the values 1, p-l,
1 +rp, and (p- l)(l + rp). These restrictions together with the relation
(3.2) restrict the degrees so heavily that they are able to reach the strong
conclusions listed in (3.7).
(12) Suppose there is a factorization in (3.6) above so that (p - l)(rp + 1) =
= (vp- l)(up+ 1). Here up- 1 =-p- 1 so that v > 1 and consequently
up+ 1 -= rpf 1 and so u-=r. Multiply out the equation, add 1 to both sides
and divide by p. This gives
This leads to the identity
rp-r+l = uvp+v-24. (3.8)
(r-u)(u+ 1) = (up+ l)(r-uv) (3.9)
which we may obtain by multiplying out the right-hand side and replacing
u2vp by u times the value of uvp from (3.8). Since r > u the left-hand side
is positive and so we may put r-uv = h where h is a positive integer.
Writing
(r-u)(u+ 1) = h(up+ 1)
(3.10)
and solving for r, we have
r = (hup+h+u2+u)l(u+1) = F(p, u, h). (3.11)
Thus the existence of further factorizations (3.6) is equivalent to expressing
r in the form (3.11) with positive h and u. We see that r =- hpu/(u + 1) > f hp,
so that for a given r, h -C 2r/p. For a given h, since u + 11 h(p - 1) there are
only a finite number of trials to be made. As g = pqw(1 +rp) and q * 2
it follows that 2rp2 -C g, 2r x g/p2 whence h -= g/p3.
(13) The general theory of modular characters of G, when g is divisible
by a power ps of a prime p higher than the first, has been used only in a
limited way. For reference see Brauer [6]. Suppose g = p”g’ where g’ 9 0
(mod p). Any element x of the finite group G has a unique expression
x = yz = zy where the order of y is a power ofp and the order z is relatively
prime to p. We call y the p-part of x. Here if the order of x is prime to p,
then y = 1. If the p-part of x is not 1 we call x p-singular, and if the p-part
of x is 1 we call x p-regular. An irreducible character x of G of degree
divisible by p” is said to be of highest type and is a p-block by itself and
vanishes for every p-singular element. An irreducible character of degree
divisible by p”-l is of defect 1 and all characters of its block have degrees
exactly divisible by ps-l.
The orthogonality relation holds :
Xe;ti1 x(x)x(y) = 0, p-parts of x and y not conjugate. (3.12)
A refinement of this, which also appears in both the Brauer-Tuan paper
[12] and the Stanton paper [30], is the following: Let p and q be different
primes and suppose that G contains no element of order pq. We quote
Lemma 2 of Stanton [30]. If G contains no elements of order pq, where
g = p%fg’, (g’, pq) = 1, and if
for all p-regular elements X, then
2 a&x) = 0, Tic B(q) a q block,
for all q-singular elements x. Furthermore
22 di(l) = 0 (mod q’), CiCB(q).
We refer to this as the principle of “block separation”. An application
is given in Example 2 of 0 5.
In Brauer-Tuan [12] it has been shown that a character of degree p’,
s 3 1, is not in the principal block B,,(p) for a simple group G.