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Poly-log diameter bounds for some families of finite groups

Authors:	Oren Dinai

Institution:	Einstein Institute of Mathematics, Edmond Safra Campus, Givat Ram, 91904 Jerusalem, Israel

Abstract:	Fix a prime and an integer with $p > m \geq 2$ . Define the family of finite groups $\displaystyle G_{n}:=SL_{m}\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)$ for $n=1,2,\ldots$ . We will prove that there exist two positive constants and such that for any and any generating set $S\subseteq G_{n}$ , $\displaystyle diam(G_{n},S)\leq C\cdot {log}^{d}(\left\vert G_{n}\right\vert)$ when $diam\left(G,S\right)$ is the diameter of the finite group with respect to the set of generators . It is defined as the maximum over $g\in G$ of the length of the shortest word in $S\cup S^{-1}$ representing . 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.

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