  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 S\'{a}rk\"{o}zy and Shkredov on the additive decomposition of the
set of quadratic residues modulo a prime.
