The Herzog-Schönheim conjecture for simple and symmetric groups

The Herzog-Schönheim conjecture states that if $H_1,\dots ,H_k$ are subgroups of a group $G$ and $x_1,\dots ,x_k$ are elements of $G$ such that $H_1{x}_1,\dots ,H_k{x}_k$ is a partition of $G$ into cosets, then two of these subgroups must have the same index. We prove this conjecture for simple groups and for symmetric groups.