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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Poly-log diameter bounds for some families of finite groups
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by Oren Dinai PDF
Proc. Amer. Math. Soc. 134 (2006), 3137-3142 Request permission

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.
References
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Additional Information
  • 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
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 3137-3142
  • MSC (2000): Primary 05C25; Secondary 05C12
  • DOI: https://doi.org/10.1090/S0002-9939-06-08384-5
  • MathSciNet review: 2231895