  Let $p$ be any prime. Let $P_n$ be a Sylow $p$-subgroup of the symmetric
group $S_n$. Let $\phi$ and $\psi$ be linear characters of $P_n$ and let $N$ be
the normaliser of $P_n$ in $S_n$. In this article we show that the inductions
of $\phi$ and $\psi$ to $S_n$ are equal if, and only if, $\phi$ and $\psi$ are
$N$--conjugate. This is an analogue for symmetric groups of a result of Navarro
for $p$-solvable groups.
