Let G 0 be a either SL n (F q ), the special linear group over the finite field with q elements, or PSL n (F q ), its projective quotient, and let Σ be a symmetric subset of G 0 , namely, if x ∈ Σ then x −1 ∈ Σ. We find a certain set R(G 0 ) of irreducible representations of G 0 whose size is at most 5, such that Σ generates G 0 if and only if |Σ| is not an eigenvalue of σ∈Σ ρ(σ) for every ρ ∈ R(G 0 ).To achieve this result, let G be either GL n (F q ) or PGL n (F q ). We consider X (G), some set of irreducible nontrivial characters of G, whose size is at most 5. We show that for every subgroup K ≤ G that does not contain G 0 , the restriction to K of at least one of the characters in X (G) contains the trivial character as an irreducible summand. We then restrict the characters to G 0 and use standard arguments about the Cayley graph of G to imply the result. In addition, we obtain slightly weaker results about the generation of symmetric subsets of G.We finish by considering S n , the symmetric group on n elements, and presenting R(S n ), a set of eight irreducible nontrivial representations of S n , such that a symmetric subset Σ ⊆ S n generates S n if and only if |Σ| is not an eigenvalue of σ∈Σ ρ(σ) for every ρ ∈ R(S n ), which is an improvement upon the previously known set of 12 irreducible nontrivial representations of S n that satisfies this condition.
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