Uniform poly-log diameter bounds for some families of finite groups

Details Uniform poly-log diameter bounds for some families of finite groups Uniform poly-log diameter bounds for some families of finite groups

2006-08-19

arxiv.org/abs/math/0608483v1

Proc. Amer. Math. Soc. 134 (2006), 3137-3142

Mathematics

Group Theory

6 pages, no figures

Scientific paper

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 $m>2$.

Affiliated with

Also associated with

No associations

Rating

Uniform poly-log diameter bounds for some families of finite groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Uniform poly-log diameter bounds for some families of finite groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Uniform poly-log diameter bounds for some families of finite groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-95407