An Upper Bound on the Number of Distinct Composition Series in Finite Groups and Extremal Behavior in Group Isomorphism

Abstract

We prove that among all finite groups of order $\le n$ (where $n \ge 4$), the number of distinct composition series is bounded above by $\prod_{i=1}^{\lfloor \log_2 n \rfloor} (2^i - 1)$, with equality holding if and only if $G$ is the elementary abelian $2$-group of order $2^\alpha$, where $\alpha = \lfloor \log_2 n \rfloor$. This bound provides a sharp, optimal upper bound for composition series enumeration in finite groups. Furthermore, we study the extremal asymptotic growth of this bound, proving it to be $n^{\frac{1}{2}\log_2 n + O(1)}$, and examine its direct implications for composition-series-based approaches to the group isomorphism problem