Are the primitive groups linearly primitive? A transitive permutation group $G \subset S_n$ is primitive if $G_1 \subset G$ is a maximal subgroup.
A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation.  
Question: Are the primitive finite groups linearly primitive?
Remark: I've checked by a GAP computation that it's true for $n=[G:G_1] \le 200$ and $\vert G \vert \le 10^4$.

Yes. The O'Nan-Scott Theorem classifies finite primitive groups into a number of different types, such as affine type, almost simple type, product type, etc. The groups in all but one of these types have a unique minimal normal subgroup $N$, which implies that they are linearly primitive, since the kernel of any unfaithful representation must contain $N$, and so any faithful representation must have an irreducible constituent whose kernel does not contain $N$ and is consequently faithful.
The groups $G$ in the remaining type have two isomorphic minimal normal subgroups $N_1$ and $N_2$, both of which are direct products of one or more isomorphic nonabelian simple groups. The smallest such example is $A_5 \times A_5$ acting on the cosets of a diagonal subgroup.
But direct products of nonabelian simple groups are linearly primitive, because the tensor product of nontrivial irreducible representations of their direct factors is a faithful irreducible representation. So $N_1 \times N_2$ has a faithful irreducible representation $\rho$ and any irreducible constituent of the induced representation $\rho^G$ is faithful for $G$.
