The Jordan-Hölder Theorem

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
exists.
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
equation
, where
is a prime , since the sequence
of decreasing positive integers

cannot continue indefinitely. We now have that is a product of primes. This proves the existence claim.

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

and

where the and are primes. Then denoting, as usual, by the cyclic group of order , we have

and

as two composition series for . But the Jordan-Hölder Theorem implies these must be equivalent; hence we must have and by suitably arranging , . Thus we have established the unique factorization theorem for positive integers as an application of the Jordan-Hölder Theroem.

David Joyner 2007-08-06