Hui Zhu
\def\zz{\bf Z} Let $A$ be a supersingular abelian variety over a finite field ${\bf k}$ which is isogenous to a power of a simple abelian variety over ${\bf k}$. Write the characteristic polynomial of the Frobenius endomorphism of $A$ relative to ${\bf k}$ as $f=g^e$ for a monic irreducible polynomial $g$ and a positive integer $e$, we show that the group of ${\bf k}$-rational points $A({\bf k})$ on $A$ is isomorphic to $(\zz/g(1)\zz)^e$ unless $A$'s simple component is of dimension $1$ or $2$, in which case we prove that $A({\bf k})$ is isomorphic to $(\zz/g(1)\zz)^a\times(\zz/{\frac{g(1)}{2}}\zz\times\zz/2\zz)^b$ for some non-negative integers $a,b$ with $a+b=e$. In particular, if the characteristic of ${\bf k}$ is $2$ or $A$ is simple of dimension greater than $2$, then $A({\bf k})\cong (\zz/g(1)\zz)^e$.