30 1. Sylow Theory

composition factors of G are the simple groups from which we might say

that G is constructed. (We mention that the finite groups for which all

composition factors are cyclic of prime order are exactly the "solvable" fi-

nite groups, which we study in more detail in Chapter 3.)

The problem of finding all nonabelian finite simple groups can be ap-

proached from two directions: construct as many simple groups as you can,

and prove that every finite nonabelian simple group appears in your list.

In particular, as part of the second of these programs, it is useful to prove

"nonsimplicity" theorems that show that groups that look different from the

known simple groups cannot, in fact, be simple. A major result of this type

from early in the 20th century is due to W. Burnside, who showed that the

order of a nonabelian simple group must have at least three different prime

divisors. (This is Burnside's classic

paqb-theorem.)

Burnside also observed that all of the then known nonabelian simple

groups had even order. He conjectured that this holds in general: that all

odd-order simple groups are cyclic of prime order. Burnside's conjecture,

which can be paraphrased as the assertion that every group of odd order

is solvable, was eventually proved by W. Feit and J. G. Thompson in the

early 1960s. (The celebrated Feit-Thompson paper, at about 250 pages,

may have been the longest published proof of a single theorem at the time.)

Since around 1960, there has been dramatic progress with both aspects of

the simple-group-classification problem, and it appears that now, in the

early years of the 21st century, the classification of finite simple groups is

complete.

So what are the nonabelian finite simple groups? A highly abbreviated

description is this. Every finite nonabelian group is either:

(1) One of the alternating groups An for n J 5,

(2) A member of one of a number of infinite families parameterized by

prime-powers q and (usually) by integers n 2, or

(3) One of 26 other "sporadic" simple groups that do not fit into types

(1) or (2).

Of the simple groups in parameterized families, the easiest to describe

are the projective special linear groups PSL(n,q), where q is a prime-

power and n 2. These are constructed as follows. Let F be the field of

order q and construct the general linear group GL{n,q) consisting of all

invertible n x n matrices over F. The special linear group SL(n, q) is the

normal subgroup of GL(n, q) consisting of those matrices with determinant

1. It is not hard to see that the center Z — Z(SX(n, q)) consists exactly

of the scalar matrices of determinant 1, and by definition, PSL{n,q) is

the factor group SL(n,q)/Z. It turns out that PSL(n,q) is simple except