This paper is concerned with products of conjugacy classes in the special linear groups SLn(F). We obtain a sufficient condition for the product of classes to cover SLn(F), depending on the shape of the rational normal form of matrices in the classes. One consequence of our result is that if the classes E-i in SLn(q) with 1 less than or equal to i less than or equal to k have the property that Pi(i=1)(k) integral(i) greater than or equal to SLn(q)(12) then Pi(i=1)(k) integral(i) = SLn(q). Our proof also gives a sharper bound on the extended covering number: any k generating conjugacy classes integral(i) in SLn(F) give Pi(i=1)(k)integral(i) = SLn(F), provided that k greater than or equal to 3n + 4. [References: 16]
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