COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

144 Marshall Hall Jr. Simple groups of order less than one million 145
A useful fact is that an algebraic conjugate of a character in the principal
block B&) is also in the principal block. Thus if in B&) there is only a
single character of a particular degree, then it is necessarily rational. For a
rational character of degree n representing a group G of order g faithfully
it has been shown by Schur [29] that the highest power of a prime p that
can divide g is p” where
s = [pJ+[&]+ . . . +[pi;el)]. (3.13)
Here the square bracket [x] denotes the integral part of x.
(14) The writer [24] has shown that if a group G has a Sylow p-subgroup
P and a normal subgroup K, then the number np of Sylow p-subgroups
in G is of the form n, = apbpcp where ap is the number of Sylow
p-subgroups in G/K, b, is the number of Sylow p-subgroups in K, and cp
is the number of Sylow p-subgroups inN&P n K)/P n K. From this it
is shown that np is the product of factors of the following two kinds:
(1) the number sp of Sylow p-subgroups in a simple group X; and (2) a
prime power q’ where q’ E 1 (mod p).
The Brauer-Reynolds results (3.7) combined with these results show
that certain numbers of the form 1 + kp cannot be the np of any finite
group: For example 15 cannot be n, in any group nor can 21 be n5 in any
group.
(15) A method attributed to Richard Brauer is quoted in the thesis of
E. L. Michaels [26]. This applies to cases in which a Sylow p-subgroup
is of order p’, r > 1. Let K be of order p’-l and the intersection of two
Sylow p-subgroups PO and PI. Then PO U PI C No(K) = H and so H
contains more than one Sylow p-subgroup, say 1 + bKp Sylow p-subgroups,
and of course every Sylow p-subgroup Pi intersecting PO in K normalizes
K. Let G contain [G : N(P)] = 1 +mp S(p)‘s and suppose that PO contains
rK conjugates of K. Counting incidences of conjugates of Kin Sylow p-subgroups
we obtain
[G : N(P)]& = (1 + bKp)[G : H]. (3.14)
Here the left-hand side says that each of 1 + mp = [G : N(P)] S(p)‘s
contains rK conjugates of K, while the right-hand side says that each of
[G : H] conjugates of Kis contained in 1 +bKp S(p)‘s. Also
mP = P c rxcbrc+ aP2,
(3.15)
K
where under conjugation by PO the remaining S(p)‘s are counted, first those
whose intersection with PO is of index p, and the rest ap2 those whose intersection
with PO is of index p2 or higher. This method is particularly useful
if the order of an S(p) is exactly p2, as the two relations (3.14) and (3.15)
are then highly restrictive.
(16) If in G there are r classes of conjugates, KI, Kz, . . . , K,, then in
the group ring R(G) over the complex field the class sums Ci = 2 x,
x E Ki, play a special role in character theory. Here
CiCj = CjCi =k~lcijkckP
where the cijk are non-negative integers. The coefficients cijk can be
expressed in terms of the characters ([14], p 316). We have
Cijk = (3.17)
Here ~9 is the value of the irreducible character xa for an element xi of the
ith class Ki and n, = x’(l) is the degree of this character. C(Xi), c(xj) are
respectively the orders of the centralizers Co(Xi), CG(xj).
A particular case of (3.17) is that in which Xi = xi = z is an involution.
If xk is of order p then, as shown in Brauer-Fowler ([8], Lemma 2A), cijk
is the number of involutions conjugate to 2 transforming xk into xc’.
In case xk is of order p and (x) is its own centralizer and q/p- I is even
this number is exactly p. As x(x) vanishes for characters not in &(p) the
equation (3.17) determines c(t) in terms of the values of x(z) and x(x)
for x in B&). In case q is odd there is no involution in N(p) and so this
number cijk must be zero. We also have Cijk = 0 if q is even but Z is not
conjugate to the involution in N(p).
4. General outline of the search. The major result of Feit and Thompson
[19] is that simple groups are of even order. Starting from the earlier results
in (9) of 0 3 it is not too difficult to show directly that there is no simple
group of odd order less than one million, and in fact a search of odd orders
less than one hundred million was completed at almost the same time
that the Feit-Thompson result was announced.
The results (2) of Gorenstein and Walter show that if g, the order of
the simple group G, is divisible by 4 but not by 8, then G is a group PS&(q).
Next suppose that g is divisible by 8 but not by 16. A Sylow 2-subgroup
S(2) is one of the five non-isomorphic groups of order 8. If S(2) is cyclic
or is the Abelian group with a basis [4,2] then S(2) is necessarily in the
center of N(2) and G has a normal 2-complement by (9). If S(2) is the
quaternion group then by the result (4) of Brauer and Suzuki G is not
simple. If S(2) is the dihedral group of order 8, then from Gorenstein and
Walter (2) G is a known group, namely PS&(q) or A,. There remains
the possibility that S(2) is the elementary Abelian group of order 8. As
S(2) is Abelian G is trivially 2-normal and the theorem of Griin ([23],
p. 215) applies, so that G has a 2-factor group isomorphic to the 2-factor
group of N(2). The automorphism group of S(2) is of order (2^3- 1)(23- 2) x
(23-22) = 168 and as N(2)/C(2) is of odd order, its order must divide 21.