By a permutation representation of a finite group $G$, we mean a homomorphism from $G$ to $S_n$, the (full) permutation group on $n$ letters.
By a linear representation of a finite group $G$, we mean a homomorphism from $G$ to $GL_n(V)$, the (full) group of invertible linear transformations from vector space $V$ to $V$ (over a field).
There are interesting applications of both the representations. For example, a group of order $2.(2n+1)$ always contains a subgroup of order $2n+1$, which can be proved easily by permutation representation.
Whereas, the groups of order $p^aq^b$ are solvable which can be proved by linear representation of groups easily; further, there are proofs without using linear representations, but they are difficult (or lengthy).
Question: Is (are) there a theorem(s) in group theory whose proof by permutation representation is easier than linear representation? I would like to know different (=more than one) such theorems if available.