Outer automorphisms of upper triangular matrices (Q1320222)

Let \(U_ n\) be the subgroup of upper triangular matrices in \(\text{GL}_ n(\mathbb{F}_ 2)\). The author calculates the outer automorphism group of \(U_ n\), which is generated by the ``flip'' (or anti-transpose), the central automorphisms and the extremal automorphisms. This implies that the classifying space \(BU_ 4\) has three dominant summands, and \(BU_ n\) has only one dominant summand for \(n \geq 5\).