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From the earliest days of group theory, researchers have been intrigued by

the question: what are the finite simple groups? Of course, the abelian

simple groups are exactly the groups of prime order, and (up to isomorphism)

there is just one group of order p for each prime p: the cyclic group of

that order. But nonabelian simple groups are comparatively rare. There

are, for example, only five numbers less than 1,000 that occur as orders of

nonabelian simple groups, and up to isomorphism, there is just one simple

group of each of these orders. (These numbers are 60, 168, 360, 504 and 660.)

There do exist numbers, however, such that there are two nonisomorphic

simple groups of that order. (The smallest such number is 8!/2 = 20,160.)

But no number is the order of three nonisomorphic simple groups.

Perhaps their rarity is one reason that nonabelian finite simple groups

have inspired such intense interest over the years. It seems quite natural to

collect rare objects and to attempt to acquire a complete collection. But a

more "practical" explanation is that a knowledge of all finite simple groups

and their properties would be a major step in understanding all finite groups.

The reason for this is that, in some sense, all finite groups are built from

simple groups.

To be more precise, suppose that G is any nontrivial finite group. By

finiteness, G has at least one maximal normal subgroup N. (We mean,

of course, a maximal proper normal subgroup, but as is customary in this

context, we have not made the word "proper" explicit.) Then by the corre-

spondence theorem, the group G/N is simple. (This is because the normal

subgroups of G/N are in natural correspondence with the normal subgroups

of G that contain TV, and there are just two of these: N and G.) Now if N

is nontrivial, we can repeat the process by choosing a maximal normal sub-

group M of N. (In general, of course, M will not be normal in G.) Because

G is finite, we see that if we continue like this, repeatedly choosing a max-

imal normal subgroup of the previously selected group, we must eventually

reach the identity subgroup. If we number our subgroups from the bottom

up, we see that we have constructed (or more accurately "chosen") a chain

of subgroups Ni such that

1 = N0 Ni • • • Nr = G

such that each of the factors Ni/Ni-i is simple for 1 i r. In this sit-

uation, the subgroups Ni are said to form a composition series for G,

and the simple groups Ni/Ni-i are the corresponding composition fac-

tors. The Jordan-Holder theorem asserts that despite the arbitrariness of

the construction of the composition series, the set of composition factors

(including multiplicities) is uniquely determined up to isomorphism. The