# Composition series and isogeny

I'm not sure this question is appropriate for this site, but it might have some connections with computational algebra.

Consider a fixed "category" $\sf{Cat}$ (in the sense of category theory, but the precise notion is not important), e.g. groups or fields. Given an object $\mathbb{S}$ of $\sf{Cat}$, we say that a "composition series" for $\mathbb{S}$ is a chain of objects $C = (\mathbb{S}_{\alpha})_{\alpha}$ indexed by some ordinal $\lambda$, such that $\mathbb{S}_0$ is an initial object and for every $\alpha < \beta$, $\mathbb{S}_{\alpha}$ is a subobject of $\mathbb{S}_{\beta}$. We may then define the length of $C$ as the ordinal $\lambda$.

We say that the structure $\mathbb{S}$ is isogenic if all maximal composition series have the same length. For instance, this is true for finite groups ordered by the normal subgroup relation (according to the Jordan-Hölder theorem). My question is: what are the structures known to have this property?

• This seems better suited to mathoverflow. – cody Apr 5 '14 at 23:14