Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
139 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Diameter of classical groups generated by transvections (2308.07086v3)

Published 14 Aug 2023 in math.GR and math.CO

Abstract: Let $G$ be a finite classical group generated by transvections, i.e., one of $\operatorname{SL}n(q)$, $\operatorname{SU}_n(q)$, $\operatorname{Sp}{2n}(q)$, or $\operatorname{O}\pm_{2n}(q)$ ($q$ even), and let $X$ be a generating set for $G$ containing at least one transvection. Building on work of Garonzi, Halasi, and Somlai, we prove that the diameter of the Cayley graph $\operatorname{Cay}(G, X)$ is bounded by $(n \log q)C$ for some constant $C$. This confirms Babai's conjecture on the diameter of finite simple groups in the case of generating sets containing a transvection. By combining this with a result of the author and Jezernik it follows that if $G$ is one of $\operatorname{SL}n(q)$, $\operatorname{SU}_n(q)$, $\operatorname{Sp}{2n}(q)$ and $X$ contains three random generators then with high probability the diameter $\operatorname{Cay}(G, X)$ is bounded by $n{O(\log q)}$. This confirms Babai's conjecture for non-orthogonal classical simple groups over small fields and three random generators.

Summary

We haven't generated a summary for this paper yet.

X Twitter Logo Streamline Icon: https://streamlinehq.com