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- W1580493345 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 $Ssubseteq G_{n}$, [ diam(G_{n},S)leq Ccdot log ^{d}(left |G_{n}right |)] when $diamleft (G,Sright )$ is the diameter of the finite group $G$ with respect to the set of generators $S$. It is defined as the maximum over $gin G$ of the length of the shortest word in $Scup S^{-1}$ 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." @default.
- W1580493345 created "2016-06-24" @default.
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- W1580493345 date "2006-06-08" @default.
- W1580493345 modified "2023-09-30" @default.
- W1580493345 title "Poly-log diameter bounds for some families of finite groups" @default.
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- W1580493345 doi "https://doi.org/10.1090/s0002-9939-06-08384-5" @default.
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