Chapter 9

Series of Subgroups

In the future, as in the past, the great ideas must be simplifying ideas.

—André Weil

If G is a finite abelian group, it can be shown that G is isomorphic to a direct product of cyclic groups (see Chapter 7). This result is an example of a structure theorem, that is, a theorem showing that every group in a suitable defined class may be constructed in a systematic way from well understood groups in the class. Such theorems are hard to come by, and the result for finite abelian groups is a stunning example. The structure of nonabelian finite groups is much more complicated.

Suppose that groups K and H are given. It is a very difficult problem to describe all groups G that have a normal subgroup K_{1} isomorphic to K such that G/K_{1} is isomorphic to H. If we could solve this extension problem, the solution would give an inductive method for constructing all finite groups. Direct and semidirect products solve this problem in very special cases. Although the general problem is far from being solved, the classes of groups that can be built up this way are of interest.

To illustrate, suppose that we use only abelian groups as building blocks. Starting with an abelian group G_{0}, we construct G_{1} ⊇ G_{0} such that G_{0} G_{1} and G_{1}/G_{0} is abelian. Next, we extend G_{1} to obtain G_{2} ⊇ G_{1} such that G_{1} G_{2} and G_{2}/G_{1} is abelian. After n steps, we have a chain

G = 389G_{n} ⊇ G_{n−1} ⊇ ...