A Group with no Composition Series

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  G be a group. A composition series of  G is a chain of normal subgroups

 1= N_0 \trianglelefteq  N_1 \trianglelefteq ... \trianglelefteq N_k = G

where  N_{i+1}/N_i is simple.

There is a theorem stating that composition series of finite groups are unique.

Theorem (Jordan-Holder). Let  G be a finite group. Consider two composition series

 1= N_0 \trianglelefteq  N_1 \trianglelefteq ... \trianglelefteq N_k = G

and

 1= M_0 \trianglelefteq  M_1 \trianglelefteq ... \trianglelefteq M_l = G

of  G. Then,  k=l and the list  \{N_1/N_0,...,N_k/N_k/N_{k-1}\} is a permutation of  \{M_1/M_0,...,M_k/M_{k-1}.\}

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:  \mathbb{Z} has no composition series. To see this, think about what the first inclusion  N \trianglelefteq \mathbb{Z} 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  \mathbb{Z}/p\mathbb{Z} will do here.

This yields  p\mathbb{Z} \trianglelefteq \mathbb{Z}. Contining forces an infinite regress

 ...\mathbb{Z}/p^3\mathbb{Z} \trianglelefteq  \mathbb{Z}/p^2\mathbb{Z} \trianglelefteq  \mathbb{Z}/p\mathbb{Z} \trianglelefteq \mathbb{Z}

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