Page "Representation theory of finite groups" Paragraph 106

id ( see for instance Exercise 6. 7 from Serre's book below ).

A consequence of this is that if χ is a non-trivial irreducible character of G such that χ ( g ) = χ ( 1 ) for some g ≠ 1 then G contains a proper non-trivial normal subgroup ( the normal subgroup is the kernel of ρ ).

Conversely, if G contains a proper non-trivial normal subgroup N, then the composition of the natural surjective group homomorphism G → G / N with the regular representation of G / N produces a representation π of G which has kernel N. Taking χ to be the character of some non-trivial subrepresentation of π, we have a character satisfying the hypothesis in the direct statement above.

Altogether, whether or not G is simple can be determined immediately by looking at the character table of G.