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Proceedings of the American Mathematical Society

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Poly-log diameter bounds for some families of finite groups

Author: Oren Dinai
Journal: Proc. Amer. Math. Soc. 134 (2006), 3137-3142
MSC (2000): Primary 05C25; Secondary 05C12
Published electronically: June 8, 2006
MathSciNet review: 2231895
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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.

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Oren Dinai
Affiliation: Einstein Institute of Mathematics, Edmond Safra Campus, Givat Ram, 91904 Jerusalem, Israel

Received by editor(s): October 26, 2004
Received by editor(s) in revised form: June 8, 2005
Published electronically: June 8, 2006
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.