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Chapter 7: Structure of Groups


The goal of a structure theory is to find the basic building blocks of the subject and then learn how they can be put together. In group theory the basic building blocks are usually taken to be the simple groups. These fit together by "stacking" one on top of the other, using factor groups.

To be more precise about this, we need to preview Definition 7.6.9. Let G be a group. A chain of subgroups
G = N0 N1 . . . Nn such that

(i) Ni is a normal subgroup in Ni-1 for i = 1,2, . . . ,n,

(ii) Ni-1 / Ni is simple for i = 1,2, . . . ,n, and

(iii) Nn = {e}
is called a composition series for G.
    The factor groups Ni-1 / Ni are called the composition factors determined by the series.

We can always find a composition series for a finite group G, by choosing N1 to be a maximal normal subgroup of G, then choosing N2 to be a maximal normal subgroup of N1, and so on. The Jordan-Hölder Theorem states that any two composition series for a finite group have the same length. Furthermore, there exists a one-to-one correspondence between composition factors of the two composition series, under which corresponding factors are isomorphic.

Unfortunately, the composition factors of a group G do not, by themselves, completely determine the group. We still need to know how to put them together. That is called the "extension problem": given a group G with a normal subgroup N such that N and G/N are simple groups, determine the possibilities for the structure of G. The most elementary possibility for G is that G can be written as the direct product N × K of N and some normal subgroup K, with K isomorphic to G/N, but there are much more interesting ways to construct G that tie the groups N and G/N together more closely.

What is known as the Hölder program for classifying all finite groups is this: first classify all finite simple groups, then solve the extension problem to determine the ways in which finite groups can be built out of simple composition factors. This attack on the structure of finite groups was begun by Otto Hölder (1859-1937) in a series of papers published during the period 1892-1895.

The simple abelian groups are precisely the cyclic groups of prime order, and groups whose simple composition factors are abelian form the class of solvable groups, which plays an important role in Galois theory. Galois himself knew that the alternating groups An are simple, for n > 5, and Camille Jordan (1838-1922) discovered several classes of simple groups defined by matrices over Zp, where p is prime. Hölder made a search for simple nonabelian groups, and showed that for order 200 or less, the only ones are A5, of order 60, and the general linear group GL3 (Z2) of all invertible 3 × 3 matrices with entries in Z2, which has order 168.

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