Unitriangular matrix group over primary finite field I have unitriangular matrix group over a field $F_q$.
It consists of the following elements:
$$E + A = \left[\begin{matrix}
    1 & a12 & ... & a1n\\
    0 & 1 & a23 & ...\\
    ... & ... & ... & ... \\
    0 & 0 & ... & 1\\
    \end{matrix}\right]$$ 
I'm trying to prove that  $(E+A)^{p^{n-1}}=E$, where $n$ is a size of matrix.
So far, I found that $(E+A)^{p}=E + A^p$  if we use Newton's binomial theorem. 
So, $(E+A)^{p^{n-1}}=E + A^{p^{n-1}}$.
But why $A^{p^{n-1}} = 0$? 
I have a hint, that for matrix $A^{p}$ first secondary diagonal is $0$, but I also don't have any idea how to prove that.

Try to use induction on $n$. Note that you can think $A = A_{n}$ as a block matrix
$$
A = A_{n} = \begin{pmatrix} A_{n-1} & v_{n-1} \\ \mathbf{0}_{1, n-1} & 0 \end{pmatrix}
$$
where $v_{n-1} \in \mathbb{F}_{p}^{n-1}$. Then it is not hard to check that 
$$
A^{m} = \begin{pmatrix} A_{n-1}^{m} & A_{n-1}^{m-1}v_{n-1} \\ \mathbf{0}_{1, n-1} & 0 \end{pmatrix}
$$
for any $m\geq 1$, so 
$$
A^{p^{n-2}} = \begin{pmatrix} A_{n-1}^{p^{n-2}} & A_{n-1}^{p^{n-2}-1}v_{n-1} \\ 0^{T} & 0 \end{pmatrix} = \begin{pmatrix} \mathbf{0}_{n-1, n-1} & A_{n-1}^{p^{n-2}-1}v_{n-1} \\ \mathbf{0}_{1, n-1} & 0 \end{pmatrix}
$$
By the way, we can apply the same argument for the same matrix with different blocks
$$
A = \begin{pmatrix} 0 & w_{n-1}^{T} \\ \mathbf{0}_{n-1, 1} & A_{n-1}' \end{pmatrix}
$$
gives us 
$$
A^{p^{n-2}} = \begin{pmatrix} 0 & w_{n-1}^{T}(A_{n-1}')^{p^{n-2}-1} \\ \mathbf{0}_{n-1, 1} & \mathbf{0}_{n-1, n-1}\end{pmatrix}
$$
This proves that $A^{p^{n-2}}$ has a form of
$$
A^{p^{n-2}} = \begin{pmatrix} \mathbf{0}_{1, n-1} & a_{1, n} \\ \mathbf{0}_{n-1, n-1} & \mathbf{0}_{n-1, 1} \end{pmatrix}
$$
and we get $A^{p^{n-1}} = (A^{p^{n-2}})^{p} = \mathbf{0}_{n, n}$. 

When $n = 3$, we have
$$
A = \begin{pmatrix} 0 & a_{12} & a_{13} \\ 0 & 0 & a_{23} \\ 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} A_{2} & v_{2} \\ \mathbf{0}_{1, 2} & 0 \end{pmatrix}
$$
where
$$
A_{2} = \begin{pmatrix} 0 & a_{12} \\ 0 & 0 \end{pmatrix}, \quad v_{2} = \begin{pmatrix} a_{13} \\ a_{23} \end{pmatrix}
$$
