  Lower bounds for the R\'enyi entropies of sums of independent random
variables taking values in cyclic groups of prime order under permutations are
established. The main ingredients of our approach are extended rearrangement
inequalities in prime cyclic groups building on Lev (2001), and notions of
stochastic ordering. Several applications are developed, including to discrete
entropy power inequalities, the Littlewood-Offord problem, and counting
solutions of certain linear systems.
