\(SL_ 3(2^ n)\) as a completion of an amalgam of type \(L_ 3(2)\)
The author shows that SL\((3,2^ n)\), \(n\neq 2\), is generated by subgroups \(X\), \(Y\), \(X \cong Y \cong S_ 4\), \(X \cap Y = B \cong D_ 8\). For this he gives concrete matrices for the elements in \(X\) and \(Y\) and shows that these generate \(\text{SL}(3,2^ n)\). The proof depends on the interesting fact that for \(n > 2\) there is an \(\alpha \in GF(2^ n)\) such that \(x^ 3 + (\alpha + 1)x^ 2 + \alpha x + 1\) is irreducible. This is not true for \(n = 2\) as \(\alpha^ 2\) is always a root. In fact the groups \(X\) and \(Y\) constructed in the proof just generate \(3A_ 6\) in \(\text{SL}(3,4)\). Finally the author shows that there are no subgroups \(X\), \(Y\) in \(SL(3,4)\), \(X\cong Y \cong S_ 4\), \(X \cap Y \cong D_ 8\), such that \(\langle X,Y\rangle = SL(3,4)\).
scientific article; zbMATH DE number 446378