 Solution

The inequality is true for ordinary addition, and also for addition in a group. For any fixed $y$, the translation $x\mapsto x+y$ is injective, so it merely relabels the probability masses in the conditional distribution. Since $X,Y$ are independent random variables,Averaging over $y$ and using conditioning reduces entropy proves entropy monotonicity under independent addition:There is no requirement that $X+Y$ remain in the original finite alphabet. The proof also gives $H(X+Y)\ge H(Y)$ by symmetry. Independence matters: for a nonconstant $X$ and $Y=-X$, the sum is constant while $H(X)>0$.