Poly-log diameter bounds for some families of finite groups

Abstract:

Fix a prime $p$ and an integer $m$ with $p > m \geq 2$. Define the family of finite groups \[ G_{n}:=SL_{m}\left (\mathbb {Z}/p^{n}\mathbb {Z}\right )\] for $n=1,2,\ldots$. We will prove that there exist two positive constants $C$ and $d$ such that for any $n$ and any generating set $S\subseteq G_{n}$, \[ diam(G_{n},S)\leq C\cdot \log ^{d}(\left |G_{n}\right |)\] when $diam\left (G,S\right )$ 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 any set of generators. The proof of this result also provides an efficient algorithm for finding such a poly-logarithmic representation of any element.