Additive decompositions of large multiplicative subgroups in finite fields

We show that a large multiplicative subgroup of a finite field ${\mathbb{F}}_q$ cannot be decomposed into $A+A$ or $A+B+C$ nontrivially. We also find new families of multiplicative subgroups that cannot be decomposed as the sum of two sets nontrivially. In particular, our results extensively generalize the results of Sárközy and Shkredov on the additive decomposition of the set of quadratic residues modulo a prime.