Uniform poly-log diameter bounds for some families of finite groups
| dc.creator | Dinai, Oren | |
| dc.date | 2006-08-19 | |
| dc.date.accessioned | 2026-07-07T07:21:56Z | |
| dc.date.available | 2026-07-07T07:21:56Z | |
| dc.description | Fix a prime $p$ and an integer $m$ with $p> m \geq 2$. Define the family of finite groups \[ G_n :=SL_m (\mathbb{Z}/p^{n}\mathbb{Z}) \] for $n=1,2,... $. We will prove that there exist two positive constants $C$ and $d$ such that for any $n$ and any generating set $S\subseteq G_{n}$, \[ diam(G_n,S) \leq C \cdot log^d (|G_n|)\] when $diam (G,S)$ is the diameter of the finite group $G$ with respect to the set of generators $S$. It is defined as the maximum over $g \in G$ of the length of the shortest word in $S \cup S^{-1}$ representing $g$. This result shows that these families of finite groups have a poly-logarithmic bound on the diameter with respect to \emph{any} set of generators. The proof of this result also provides a efficient algorithm for finding such a poly-logarithmic representation of any element. In addition it shows that the power $d$ in the $log$ bound can be arbitrary close to 3 for $m=2$ and arbitrary close to 4 for $m>2$. | |
| dc.description | 6 pages, no figures | |
| dc.identifier | https://arxiv.org/abs/math/0608483 | |
| dc.identifier | http://arxiv.org/abs/math/0608483 | |
| dc.identifier | Proc. Amer. Math. Soc. 134 (2006), 3137-3142 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/115481 | |
| dc.subject | Group Theory | |
| dc.subject | Combinatorics | |
| dc.subject | 05C25; 05C12 | |
| dc.title | Uniform poly-log diameter bounds for some families of finite groups | |
| dc.type | text |