On the product decomposition conjecture for finite simple groups. (English) Zbl 1355.20013
The authors make some advance towards a conjecture of M. W. Liebeck et al. [Bull. Lond. Math. Soc. 44, No. 3, 469–472 (2012; Zbl 1250.20018)], which says that for any subset \(S\) of a simple group \(G\) with \(|S|>1\), \(G\) is a product of \(O( \log |G| / \log |S|)\) conjugates of \(S\). This is proved if \(G\) is of Lie type, with the constant in the \(O\) depending on its rank.
They also give an estimate for the rate of increase of product sets. Given a set \(S\), for some \(g\in G\) we have \(|SgS| \geq |S|^{1+\varepsilon}\) unless already \(S^3=G\), \(\varepsilon\) depending on the rank. For normal subsets, the dependence can be eliminated.
They also give an estimate for the rate of increase of product sets. Given a set \(S\), for some \(g\in G\) we have \(|SgS| \geq |S|^{1+\varepsilon}\) unless already \(S^3=G\), \(\varepsilon\) depending on the rank. For normal subsets, the dependence can be eliminated.
Reviewer: Imre Z. Ruzsa (Budapest)
MSC:
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20D40 | Products of subgroups of abstract finite groups |
| 20G40 | Linear algebraic groups over finite fields |
Citations:
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