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

• URL: http://arxiv.org/abs/2511.19494v1
• Date: Sun, 23 Nov 2025 12:30:46 GMT
• Title: Probabilistic Bounds on the Number of Elements to Generate Finite Nilpotent Groups and Their Applications
• Authors: Ziyuan Dong, Xiang Fan, Tengxun Zhong, Daowen Qiu,
• Abstract summary: We prove that $varphi_k(G) ge 1 - $ if $k ge operatornamerank(G) + lceil log(2/) rceil$ (a bound based on the group rank) or if $k ge operatornamelen(G) + lceil log (1/) rceil$ (a bound based on the group chain length)
• Score: 4.250782756734906
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This work establishes a new probabilistic bound on the number of elements to generate finite nilpotent groups. Let $\varphi_k(G)$ denote the probability that $k$ random elements generate a finite nilpotent group $G$. For any $0 < ε< 1$, we prove that $\varphi_k(G) \ge 1 - ε$ if $k \ge \operatorname{rank}(G) + \lceil \log_2(2/ε) \rceil$ (a bound based on the group rank) or if $k \ge \operatorname{len}(G) + \lceil \log_2(1/ε) \rceil$ (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 $k \ge \lceil \log_2 |G| + \log_2(1/ε) \rceil + 2$. 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.

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