  Let $p$ be a prime number and let $k$ be an algebraically closed field of
characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative
group scheme which arises as the kernel of $p$ on a $p$-divisible
(Barsotti--Tate) group. We compare three classifications of $BT_1$ group
schemes, due in large part to Kraft, Ekedahl, and Oort, and defined using
words, canonical filtrations, and permutations. Using this comparison, we
determine the Ekedahl--Oort types of Fermat quotient curves and we compute four
invariants of the $p$-torsion group schemes of these curves.
