# Normal Subgroup and upper triangular matrices

$$B_n$$ is the subgroup of upper triangular matrices in $$GL_n(\mathbb{R})$$. $$T_n$$ is the group of diagonal matrices in $$GL_n(\mathbb{R})$$, $$U_n \subset B_n$$ is the subgroup of matrices whose diagonal entries are 1.

I was trying to prove that $$B_n$$ is a semidirect product of $$U_n$$ and $$T_n$$. There are similar questions on the site, but none of them address my question.

My approach is to prove:

1. $$U_n \cap T_n = \{e\}$$ which is easy to see
2. $$U_n \triangleleft B_n$$
3. $$B_n = U_nT_n$$

I have problems with 2 and 3. For 2, I used the definition of the normal subgroup (conjugation) and tested the $$2 \times 2$$ case (Normal Subgroup of T (upper triangular matrices under multiplication)) but I was unable to prove it in general, and I don't know how to do 3.

Any help will be appreciated.

3. For a given upper triangular matrix with diagonal entries $$d_i$$, multiply it by $$\mathrm{diag}(1/d_1,\dots, 1/d_n)$$ to obtain a matrix in $$U_n$$.