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.