Let $GL_n \Bbb R$ be the set of all invertible matrices of size n. Let $U_n$ be the set of upper triangular matrices with $1$'s on the diagonal and $D_n$ be the set of diagonal matrices with non-zero entries. Let G be the subgroup generated by $U_n$ & $D_n$. Prove that $G $ is the semidirect product of $U_n$ & $D_n$.
It is clear that $U_n$ is normal in $G$. But what is the order of Aut($U_n$)? and What will be the map? Please help.