There is an exercise in the book "An Introduction to the group theory by J.J. Rose" which can also be found as a proposition in "Abstract algebra by T. Hungerford":
Every finite group has a composition series $^*$.
Now I am doing the exercise $5.9$ of the first above book:
An abelian group has a composition series iff it is finite.
Give an example of an infinite group which has a composition series.
About 1. : Since $(*)$; one side can be carried out. For other side; what would be happened if we assumed the group was infinite? In fact, if an abelian group is infinite; it cannot have a composition series with finite length? Is this our contradiction? I see this by considering $\mathbb Z_{p^\infty}$ but cannot see the right way. Thanks.
Best Answer
A non-trivial simple Abelian group is cyclic of prime order. A composition series must have finite length. This should suffice.