If a group (or module) admits a finite composition series,is it possible that it has normal series (submodule series) of arbitrary length?

Question

If a group (or module) admits a finite composition series,is it possible that it has normal series (submodule series) of arbitrary length?

I think it is impossible, but I can't find a proof in website, could you help me? Thanks.

Answer 1

Magic words are "Schreier refinement Theorem".

Answer 2

For modules, I would say the easiest idea would be to use the characterization of modules with finite composition series as those which are both Artinian and Noetherian. You can find a proof of this, for example, in Martin Isaacs' Graduate algebra Theorem 11.3 [google books], or here on page 3.

I expect the same should be true for groups but I really can't find an analogous statement for groups which would work the same way (although it clearly holds for abelian groups, which are just $\mathbb Z$ modules.)

The only hint I had in the positive direction was this exercise in the second link above:

But I'm reluctant to declare that as the final word since I couldn't find it elsewhere and haven't verified it.