Entropy of a sum of independent finite-group variables
= Entropy of a sum of independent finite-group variables
If <independent random variables> $X,Y$ take values in a <finite additive group>, then
$$
H(X+Y)\geq\max\{H(X),H(Y)\}.
$$
Indeed, $H(X+Y)\geq H(X+Y\mid Y)=H(X)$, and symmetrically for $Y$. Independence is essential: taking $Y=-X$ makes the sum constant.