Wednesday, 24 January 2018
Okan Özkan: Products of commutators in finite simple groups
Since every non-abelian simple group is perfect, it is only natural to ask whether there is a natural number \(j\) such that every element of a finite non-abelian simple group is a product of \(j\) commutators. Wilson proved that indeed there is such a \(j\).In this talk I will first prove (in parts) a model theoretic result of Point, namely that \(X(K)\) is isomorphic to the ultraproduct of the \( X(K_i)\) where \(X\) is a Lie type, the \( K_i \) form a family of finite fields and \(K\) is the ultraproduct of the \(K_i\) (we use a non-principal ultrafilter on the natural numbers for the ultraproducts). Using Point's result I will show that Wilson's theorem holds for groups of the same Lie type, i.e. \(j=j(X)\) is depending on the Lie type. Together with a number of other ingredients one can finally prove Wilson's result.