Growth and generation in $\mathrm{SL}_2(\mathbb{Z}/p \mathbb{Z})$
Annals of mathematics, Tome 167 (2008) no. 2, pp. 601-623
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We show that every subset of $\mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})$ grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators $A$ of $\mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})$, every element of $\mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})$ can be expressed as a product of at most $O((\log p)^c)$ elements of $A \cup A^{-1}$, where $c$ and the implied constant are absolute.
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author = {Harald A. Helfgott},
title = {Growth and generation in $\mathrm{SL}_2(\mathbb{Z}/p \mathbb{Z})$},
journal = {Annals of mathematics},
pages = {601--623},
publisher = {mathdoc},
volume = {167},
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year = {2008},
doi = {10.4007/annals.2008.167.601},
mrnumber = {2415382},
zbl = {1213.20045},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.167.601/}
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Harald A. Helfgott. Growth and generation in $\mathrm{SL}_2(\mathbb{Z}/p \mathbb{Z})$. Annals of mathematics, Tome 167 (2008) no. 2, pp. 601-623. doi: 10.4007/annals.2008.167.601
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