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
Left/Right Noetherian and Left/Right Artinian Rings
Counterexamples in Upper Triangles Matrix Rings
The ring is left Noetherian. However, it is not right Noetherian, left Artinian or right Artinian.
Similarly, the ring is left Artinian and left Noetherian, but not right Artinian or right Noetherian.
The details are worked out here under “CIA: Some Upper Triangular Matrix Rings and Noetherian/Artinian Hypotheses.”
Noetherian Ring with Infinite Krull Dimension
This one is due to Nagata. Consider a polynomial ring in countably infinitely many indeterminates over a field Let, for each , denote the prime ideal . Then, define the (multiplicative) subset . The localization will be our counterexample.
We claim that is Noetherian. To see this, consider a prime ideal . First, each is maximal and each is contained in only finitely many . So, is contained in only finitely many . For each , is finitely generated. Thus, we can obtain a finite list of elements of that generate each Note that, by Hilbert’s basis theorem, is Noetherian. Let Observe that for all but finitely many maximal ideals but the are not any of the in which is contained. Moreover, for all . Therefore, gives in each localization and so is finitely generated. As all prime ideals of are finitely generated, it is Noetherian.
Lastly, for each there is a chain of prime ideals. The supremum then of the lengths of chains of prime ideals in is infinite e.g. has infinite Krull dimension.