In order to state the main theorem of this section, we first need two definitions.
On the basis of Corollary 8.3.3 another way of characterizing a maximal normal subgroup is as follows: is a maximal normal subgroup of if and only if is simple. (See the exercises below.)
We now state our last definition.
For example, in the case of the previously given normal series for in (11.8), only the first is a composition series for . A composition series for would be: . Note that would not be a composition series for (Why?). Unlike the case of normal series, it is possible that an arbitrary group does not have a composition series (see exercise 1 for this section) or even if it does have one a subgroup of it may not have one. Of course, a finite group does have a composition series.
We now consider the case in which a group, , does have a composition series, and we prove the following important theorem.
Proof: Suppose we are given two composition series. Applying Schreier's refinement theorem (Theorem 11.2.2), we get that the two composition series have equivalent refinements. But the only refinement of a composition series is one obtained by introducing repetitions. If in the 1-1 correspondnece between the factors of these refinements, the paired factors equal to are disregarded (i.e., if we drop the repetitions), we get clearly that the original composition series are equivalent.
It was mentioned in the introduction to Chapter 6 that the simple groups are important because ``they play a role in finite group theory somewhat analogous to that of the primes in number theory.'' In particular, an arbitrary finite group, , can be broken down into simple components. These uniquely determined simple components are, according to the Jordan-Hölder, the factors of a composition series for .
We close by giving an application of this theorem. In particular, we use the Jordan-Hölder Theorem to prove the uniqueness part of the Fundamental Theorem of Arithmetic. The Fundamental Theorem of Arithmetic states that every positive integer not equal to a prime can be factored uniquely (up to order) into a product of primes.
First, we claim that such a factorization
Indeed, suppose is composite (i.e.,
and is not a prime). Then an easy induction shows
that has a prime divisor and we can write ,
where is an integer satisfying
. If is prime, the claim holds.
Otherwise, has a prime factor , and
where is an integer.
Continuing in this fashion, we must come to an
is a prime , since the sequence
of decreasing positive integers
On the basis of the Jordan-Hölder Theorem,
we can easily show the other part of the Fundamental
Theorem of Arithmetic, i.e., apart from order
of the factors, the representation of as product of primes is
unique. To do this suppose that