Existence of simple non-cyclic abelian varieties over arbitrary finite fields and of a given dimension $g>1$

Vlăduţ characterized in 1999 the set of finite fields k such that all elliptic curves defined over k have a cyclic group of rational points. Under the conjecture of infinitely many Mersenne primes, this set is infinite. In these notes we prove that there is no a finite field k such that all the simple abelian varieties defined over k of dimension $g>1$ have a cyclic group of rational points.