Classification of abelian subgroups of maximum order in unipotent upper-triangular matrix groups

Statement

Let $p$ be a prime number, and let $q = p^r$ be a power of $p$. Let $U(n,q)$ denote the group of upper-triangular unipotent $n \times n$ matrices over the field of $q$ elements. Note that $U(n,q)$ is a $p$-Sylow subgroup of the general linear group $GL(n,q)$.

Then the abelian subgroups of maximum order in $U(n,q)$ are given as follows:

• For odd $n$, there are two such groups, both of them elementary Abelian of order $q^{(n^2 - 1)/4}$. The first group is given as the group of all upper-triangular matrices with $1$s on the diagonal, and zeroes elsewhere except in the top right $(n-1)/2 \times (n+1)/2$ rectangle. The second group is given as the group of all upper-triangular matrices with $1$s on the diagonal, and zeroes elsewhere except in the top right $(n+1)/2 \times (n-1)/2$ rectangle. Both these subgroups are normal: they are related via an outer automorphism which is the composite of conjugation by the antidiagonal matrix and the transpose-inverse map.
• For even $n$, there is exactly one Abelian subgroup of maximum order. This is elementary Abelian of order $q^{n^2/4}$. It is given as the group of upper-triangular matrices with $1$s on the diagonal and zeroes elsewhere except in the top right $n \times n$ square.