If $G \subset GL_n(\mathbb R)$ is a group consisting of upper-triangular matrices under matrix multiplication and $H$ is subgroup of $G$, but with $1$s on diagonal, prove that $H$ is normal in $G$.
The definition $ghg^{-1} \in H$, for every $g$ and $h$ seems a bit tedious, especially because the matrices are $n \times n$ , for some natural number n.
I tried finding a homomorphism such that $H$ is its kernel, which would imply it's normal, but I failed to do so.
Any help?