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

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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 1s 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 1s 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 1s on the diagonal and zeroes elsewhere except in the top right n \times n square.

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