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arXiv · 2609.26307

Word Length and Diameter in Permutation Groups

Markus Lohrey·Alexander Thumm
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Abstract

The input for the binary diameter problem consists of explicitly represented permutations generating a finite group $G$ and a binary-encoded nonnegative integer $k$. The question is whether every element of $G$ is a product of at most $k$ input generators. For the binary length problem, the input contains in addition a permutation $g \in G$ and it is asked whether $g$ is a product of at most $k$ input generators. We prove that the binary diameter problem is PSPACE-complete. When restricted to $2$-step nilpotent groups, the binary diameter problem is shown to be complete for $\mathsf{Π_2^P}$, whereas the binary length problem is shown to be NP-complete. Without the restriction to $2$-step nilpotent groups, the binary length problem is PSPACE-complete by a result of Jerrum.

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Markus Lohrey, Alexander Thumm. 2026-09-22. Word Length and Diameter in Permutation Groups. https://arxiv.org/abs/2609.26307

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