Chain of normal domains for $GL_n(\mathbb{F}_p)$

2017-12-03 03:05:11

There are $$p^\frac{n\cdot(n-1)}{2}\cdot(p^n-1)\cdot\ldots(p-1)$$ elements in $GL_n(\mathbb{F}_p)$, so we consider a subgroup $H$ of order $p^\frac{n\cdot(n-1)}{2}$ as a $p$-sylow. One finds that the upperdiagonal matrices with diagonal elements equal to $1$ is closed under matrix multiplication with order $p^\frac{n\cdot(n-1)}{2}$.

How can one find now a chain of normal domains $$ H_0=\{0\}\subset H_1\subset\ldots\subset H_k=H,$$ such that $H_i\lhd H$ and $H_{i}/H_{i-1}\cong\mathbb{Z}/p\mathbb{Z}$ for all $1\le i\le k$?