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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.

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The "standard" proofs of the simplicity of the finite classical groups like ${\rm PSL}(n,q)$ make heavy use of permutation representations and the general theory of permutation groups. The final step in the proofs uses Iwasawa’s Theorem, which is criterion for a primitive permutation group to be simple.

Of course the definition of these groups is in terms of matrix groups over finite fields, and so inevitably parts of the proof involve examining the action of the group on the underlying vector space. For the linear groups, you show that the action on $1$-dimensional subspaces is $2$-transitive, and that the point stabilizer has an abelian normal subgroup with certain properties that you need to apply Iwasaw's Theorem.

But even so, I have never thought of these proofs as involving what we usually call Represnetation Theory.

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