COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

140 Marshall Hall Jr. Simple groups of order less than one million 141
group has an Abelian Sylow p-subgroup S(p) = PO there must be a conjugate
PI with PC, 0 PI = 1,
The Brauer theory of modular characters has played a major role in this
search, in particular the theory for groups whose order is divisible by
exactly the first power of a prime. Let p be a prime and suppose that g, the
order of the group G, is divisible by exactly the first power of p. We write
g = pqw(l+rp), p-l = qt. (3.1)
Here G has 1 frp Sylow p-subgroups S(p) and the order of the normalizer
N(p) of a Sylow p-subgroup is IN(p)/ = pqw. The centralizer C(p) of S(p) is
of order pw and C(p) = S(p)X V where V = V(p) is of order w. N(p)/@)
is cyclic of order q where qJp - 1 is the order of the group of automorphisms
of S(p) induced by N(p). By a classical theorem of Burnside’s ([14], p. 203)
if we had q = 1, G would have a normal p-complement and so we have
q =- 1 for a simple group.
(6) Brauer [3]. The principal block of characters Be(p) contains q ordinary
characters x1, x2, . . . , xg where x1 is the trivial character of degree fr = 1
and xi( 1) = J;:, i = 2, . . . , q, and a family of t = (p - 1)/q exceptional characters
xi which are p-conjugate and are all of the same degree x&l) = fo,
j= 1, . . ., t. For the ordinary characters there is a sign 6i = + 1 such that
SiJ = 1 (mod p) and for the exceptional character there is a sign 60 = f 1
such that S,& = - q (mod p). If q = p - 1 the single exceptional character
is not distinguishable from the ordinary character. Also the degrees satisfy
the relations
1+S&+ . . . +~qfq+~ofo = 0,
(3.2a)
jY jq(l+rp). u = 0, 1, . . ., q.
If u is a generator of S(p) then
xi(U)=6iy i=l,..., q, (3.2b)
&4) = - 80(&” + ES” + . . . + &“‘-‘q, 2, = G),
where e is a primitive ith root of unity, s is a primitive solution of sQ z 1
(mod p) and ZJ = v(j) ranges over t values such that vsi gives a full set of a
non-zero residues modulo p. These values of the exceptional characters are
the Gauss periods of cyclotomy. If w = 1 then B&) is the only block containing
characters of degrees not divisible by p. If w =- 1 there are further
non-principal blocks of characters of degrees not divisible by p and Brauer
gives relations similar to (3.2a, b) for these blocks. A character x of degree
divisible by p is a block by itself and vanishes for everyp-singular element
(an element of order divisible by p). No simple group of order divisible by p
has an (irreducible) character (except the trivial one) of degree less than
&- 1) and if it does have an irreducible character of degree i(p- 1) then
it is isomorphic to PSL&). This has been shown to be true by Feit and
Thompson [20] even if a higher power of p than the first divides the order of
the group. More recently Feit [18] has extended this to show the same conclusion
if G has a character of degree less than p - 1. Everett Dade [16] has
extended Brauer’s work to cover the case in which a Sylow subgroup is
cyclic.
(7) It was shown by Brauer [3] that, for groups divisible by only the first
power of a primep, the characters of a block may be associated with a
tree. Each ordinary character is a vertex and each modular character is an
arc. An ordinary character (treating a family of p-conjugate exceptional
characters as a single character) decomposes as the sum of the modular characters
which are arcs with an end at the vertex for the ordinary character.
A modular character appears as a constituent of exactly two ordinary characters
and in each of these with multiplicity one. If xi and Xk are the two
ends of an arc then Xi+Xk is a modular indecomposable and vanishes for
everyp-singular element, and so in particular if 6i and 6, are the corresponding
signs 6if fik = 0. H. F. Tuan [34] has refined this for the principal block
and has shown that the real characters (characters real for every element)
in the tree form a stem which may be drawn in a straight line, and the tree is
symmetric with respect to this stem with complex conjugacy interchanging
the remaining vertices and arcs.
(8) Brauer and Tuan ([12], Lemma 1) showed that for x a character in the
principal p block B&), in the notation of (5), then the restriction of x to V,
xl V, has the form
x / V = moofp0 (3.3)
where ,a0 is the identity character of V and 0 is some character of V possibly
reducible. In private communication to me, Leonard Scott has generalized
this to characters x of a non-principal block Bj(p). In this case the formula
becomes
x 1 V = miSNlfpO
i=l
where TN*, i = 1, . . . , s, are the irreducible characters of V conjugate in
N(p) associated with the block Bj(p). In case the character x has degree divisible
by p, a simple consequence of the fact that x vanishes for p-singular
elements yields
x j v = pe. (3.5)
The formula (3.4) gives for an x E V the determinantal relation
det (x(x)) = det (e(x)) p, so that if 8 is of degree 1 and G is simple then
det e(x) = 1, whence e(x) = 1 and x(x) = x(1) is the identity matrix, a situation
impossible for a simple group. In particular with g = pqw( 1 + rp) for a
simple group G, if Bob) contains a character of degree less than 2p, then
w = 1. This is also proved and used by Stanton [30] in his study of the
Mathieu groups. To avoid confusion with other references to Brauer and
Tuan we shall call this the Stanton condition.