A Group with no Composition Series
Spookiness in Group Theory
The Jordan-Holder theorem is a group-theoretic analogue of the uniqueness of prime factorizations. A “factorization” of a (finite) group is called a composition series.
Definition. Let be a group. A composition series of is a chain of normal subgroups
where is simple.
There is a theorem stating that composition series of finite groups are unique.
Theorem (Jordan-Holder). Let be a finite group. Consider two composition series
of Then, and the list is a permutation of
So, a composition series tells us a lot about a group. Therefore, we like them. Lucky for us then, there is a theorem stating that all finite groups admit a composition series. But, this is not true for infinite groups.
This bring us to our counterexample: has no composition series. To see this, think about what the first inclusion must be (hint: the quotient must be a finite simple group). Those of us reading this post any time after 2004 have the privilege of being able to say that there is a list of all finite simple groups; only will do here.
This yields Contining forces an infinite regress