  Let $G_0$ be a either $SL_n(\mathbb{F}_q)$, the special linear group over the
finite field with $q$ elements, or $PSL_n(\mathbb{F}_q)$, its projective
quotient, and let $\Sigma$ be a symmetric subset of $G_0$, namely, if $x \in
\Sigma$ then $x^{-1} \in \Sigma$. We find a certain set $\mathcal{R}(G_0)$ of
irreducible representations of $G_0$ whose size is at most $5$, such that
$\Sigma$ generates $G_0$ if and only if $\left\lvert\Sigma\right\rvert$ is not
an eigenvalue of ${\sum_{\sigma \in \Sigma} \rho(\sigma)}$ for every $\rho \in
\mathcal{R}(G_0)$.
  To achieve this result, let $G$ be either $GL_n(\mathbb{F}_q)$ or
$PGL_n(\mathbb{F}_q)$. We consider $\mathcal{X}(G)$, some set of irreducible
nontrivial characters of $G$, whose size is at most $5$. We show that for every
subgroup $K \le G$ that does not contain $G_0$, the restriction to $K$ of at
least one of the characters in $\mathcal{X}(G)$ 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 $\mathcal{R}(S_{n})$, a set of eight irreducible nontrivial
representations of $S_{n}$, such that a symmetric subset $\Sigma \subseteq
S_{n}$ generates $S_{n}$ if and only if $\left\lvert\Sigma\right\rvert$ is not
an eigenvalue of ${\sum_{\sigma \in \Sigma} \rho(\sigma)}$ for every $\rho \in
\mathcal{R}(S_{n})$, which is an improvement upon the previously known set of
$12$ irreducible nontrivial representations of $S_{n}$ that satisfies this
condition.
