Difference algebraic subgroups of commutative algebraic groups over finitefields

We study the question of which torsion subgroups of commutative algebraic g roups over finite fields are contained in modular difference algebraic grou ps for some choice of a field automorphism. We show that if G is a simple c ommutative algebraic group over a finite field of characteristic p, l is a prime different from p, and for some difference closed field (K, sigma) the l-primary torsion of G(K) is contained in a modular group definable in (K, sigma), then it is contained in a group of the form {x is an element of G( K) : sigma(x) = [a](x)} with a is an element of N\p(N). We show that no suc h modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Bo xall.