Let $G$ be a primitive group acting on a set $\Omega$ with $n$ elements. By Cameron/Liebeck (essentially a consequence of the Classification + O'Nan-Scott), there are two possibilities:


(a) $G$ has a subgroup of index $\leq n$ isomorphic to an alternating group, 


(b) $G$ is of size $\leq n^{O(\log n)}$. 


In case (a), $G$ cannot have a composition series $$\{e\} = H\_0 \triangleleft H\_1 \triangleleft \dotsc \triangleleft H\_{\ell} = G$$ of length $\ell$ greater than $O(\log n)$.


In case (b), the trivial bound on the length $\ell$ of a composition series is $\ell = O((\log n)^2)$. In fact, the trivial bound on the length of a chain of subgroups $\{e\} = H\_0 \lneq H\_1 \lneq \dotsc \lneq H\_{\ell'} = G$ is also $O((\log n)^2)$.


Questions:


1. (The question I asked at first.) In case (b), can one give a better bound on $\ell'$ than $O((\log n)^2)$? (Answer: no; see below.)
2. (The question I meant to ask.) In case (b), can one give a better on bound on $\ell$ than $O((\log n)^2)$? Perhaps $O(\log n)$? (Answer: yes; see below.)



Length of composition series in a primitive group
To answer your new question, Theorem 1.3 of [this arXiv paper](https://arxiv.org/abs/1712.05520) proves that the composition length $c(G)$ of primitive $G \le S\_n$ satisfies
$$c(G) \le \frac{8}{3} \log\_2 n - \frac{4}{3}$$
and also identifies the examples in which equality holds.


