Probabilistic Bounds on the Number of Elements to Generate Finite Nilpotent Groups and Their Applications

Abstract

This work establishes a new probabilistic bound on the number of elements to generate finite nilpotent groups. Let $\ddot{I}{†}_k(G)$ denote the probability that $k$ random elements generate a finite nilpotent group $G$. For any $0<\hat{I}\mu <1$, we prove that $\ddot{I}{†}_k(G)\ge 1-\hat{I}\mu$ if $k\ge \mathrm{rank}(G)+\lceil {\log }_2(2/\hat{I}\mu )\rceil$ (a bound based on the group rank) or if $k\ge \mathrm{len}(G)+\lceil {\log }_2(1/\hat{I}\mu )\rceil$ (a bound based on the group chain length). Moreover, these bounds are shown to be nearly tight. Both bounds sharpen the previously known requirement of $k\ge \lceil {\log }_2|G|+{\log }_2(1/\hat{I}\mu )\rceil +2$. Our results provide a foundational tool for analyzing probabilistic algorithms, enabling a better estimation of the iteration count for the finite Abelian hidden subgroup problem (AHSP) standard quantum algorithm and a reduction in the circuit repetitions required by Regev's factoring algorithm.