[Math] Showing that a cyclic group of prime power order has only 1 composition series

abstract-algebrafinite-groups

I am trying to show that a cyclic group of prime power order has only 1 composition series. Is the following correct?

Let $G=C_{p^n}$. Then as cyclic groups are abelian we have that there is a subgroup of every order dividing $p^n$, which is $1,p,p^2,…$, then these are all normal (as $G$ is abelian) and are cyclic and so are unique up to isomorphism.

So if we start of with any subnormal series $\displaystyle{\{1\}\triangleleft C_{p^i} \triangleleft C_{p^n}}$ then we can always refine it so that it is:

$\{1\}\triangleleft C_p\triangleleft C_{p^2}\triangleleft …..\triangleleft C_{p^m}$

Best Answer

That looks correct. The facts you need are:

  1. A (finite) cyclic group has at most one subgroup of a given order (or, to put it differently, any cyclic group has at most one subgroup of a given index).
  2. A quotient of a cyclic group is cyclic.
  3. A subgroup of a cyclic group is cyclic.
  4. A nontrivial cyclic group is simple iff it is of prime order.
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