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 =
N_{0}
N_{1}
. . .
N_{n} such that

**(i)**N_{i}is a normal subgroup in N_{i-1}for i = 1,2, . . . ,n,**(ii)**N_{i-1}/ N_{i}is simple for i = 1,2, . . . ,n, and**(iii)**N_{n}= {e}

The factor groups N

We can always find a composition series for a finite group G,
by choosing N_{1} to be a maximal normal subgroup of G,
then choosing N_{2} to be a maximal normal subgroup of N_{1},
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
A_{n} are simple, for n > 5,
and Camille Jordan (1838-1922) discovered several classes
of simple groups defined by matrices over **Z**_{p},
where p is prime.
Hölder made a search for simple nonabelian groups,
and showed that for order 200 or less,
the only ones are A_{5}, of order 60, and the
general linear group
GL_{3} (**Z**_{2}) of all invertible 3 × 3 matrices
with entries in **Z**_{2}, which has order 168.

Forward to §7.1 | Up | Table of Contents | About this document