= Solution
The simple <modules> are indexed by the <p-regular partitions> $\lambda$ of $n$, meaning no part is repeated $p$ or more times. Equip the <tabloid> <permutation module> with the <bilinear form> invariant under the <symmetric group> making its <tabloid> <basis> orthonormal. The construction is
$$
D^\lambda=S^\lambda/(S^\lambda\cap(S^\lambda)^\perp).
$$
The standard modular Specht theorem says this quotient is nonzero precisely for p-regular $\lambda$, is then simple, and these quotients give all simple <modules>.
To verify distinctness of labels, use the standard dominance triangularity theorem: if $D^\nu$ occurs as a composition factor of $S^\lambda$, then $\nu\ge\lambda$; for a p-regular label the diagonal multiplicity is one. If $D^\lambda\cong D^\mu$, then $D^\lambda$ occurs in $S^\mu$, so $\lambda\ge\mu$, and the symmetric argument gives $\mu\ge\lambda$. Antisymmetry of the <dominance order on partitions> yields $\boxed{\lambda=\mu}$. These are the general representation-theoretic results allowed in this part.
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