Classification of abelian subgroups of maximum order in unipotent upper-triangular matrix groups
Statement
Let be a prime number, and let be a power of . Let denote the group of upper-triangular unipotent matrices over the field of elements. Note that is a -Sylow subgroup of the general linear group .
Then the abelian subgroups of maximum order in are given as follows:
- For odd , there are two such groups, both of them elementary Abelian of order . The first group is given as the group of all upper-triangular matrices with s on the diagonal, and zeroes elsewhere except in the top right rectangle. The second group is given as the group of all upper-triangular matrices with s on the diagonal, and zeroes elsewhere except in the top right 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 , there is exactly one Abelian subgroup of maximum order. This is elementary Abelian of order . It is given as the group of upper-triangular matrices with s on the diagonal and zeroes elsewhere except in the top right square.
References
- Abelian p-subgroups of the general linear group by J. T. Goozeff, Journal of the Australian Mathematical Society, Volume 11, Page 257 - 259(Year 1970): This paper finds the Abelian subgroups of maximum order in the group of upper-triangular unipotent matrices over a finite field .^{}^{More info}