IE 31
when n = 2 and g is 2 or 3. (We say more about the groups PSL(n, q) in
Chapter 7, and we prove their simplicity in Chapter 8, where we also prove
that the alternating groups An are simple for n 5.)
In fact, all of the simple groups with order less than 1,000 are of the
form PSX(2,g), where q is one of 5, 7, 8, 9 or 11, and the corresponding
group orders are 60, 168, 504, 360 and 660. The unique (up to isomor-
phism) simple group PSX(2, 5) of order 60 has two other realizations: it is
isomorphic to PSX(2,4) and also to the alternating group A$. The simple
groups PSX(2, 7) of order 168 and PSX(2,9) of order 360 also have multiple
realizations: the first of these is isomorphic to PSX(3, 2) and the second is
isomorphic to ^6-
The smallest of the 26 sporadic simple groups is the small Mathieu group,
denoted Mn, of order 7,920; the largest is the Fischer-Griess "monster" of
order
246.320-59-76-ll2-17-19-23-29-31-41.47-59-71
=
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000.
For the remainder of this section, we discuss nonsimplicity theorems of
the form: "if the order of G is ..., then G cannot be simple". (Of course, both
Burnside's
paqb-theorem
and the Feit-Thompson odd-order theorem are of
this type.) A very much more elementary result of this form is immediate
from the fact that a nontrivial p-group always has a nontrivial center. If
\G\ is divisible by only one prime, it follows that G cannot be simple unless
Z(G) = G, and in this case, G is abelian, and so it must be cyclic of prime
order.
Burnside's
paqb-theorem,
asserts that the order of a simple group cannot
have exactly two prime divisors. His beautiful (and short) proof is not
elementary since it uses character theory (which we do not discuss in this
book). It took about 50 years before a purely group-theoretic (and harder)
proof of Burnside's theorem was found by Goldschmidt, Matsuyama and
Bender, using powerful techniques of Thompson, Glauberman and others.
These techniques were developed for other purposes, and eventually they led
to the full classification of finite simple groups, but as a test of their power,
it seemed reasonable to see if they would yield a direct proof of Burnside's
theorem. Indeed they did, and one of the goals of this book is to develop
enough group theory so that we can present a somewhat simplified version
of the Goldschmidt-Matsuyama-Bender proof of Burnside's theorem. (This
proof appears in Chapter 7.)
For now, however, we use Sylow theory (and a few tricks) to prove some
much more elementary nonsimplicity theorems.
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