Central implies image under every irreducible representation is scalar

Statement
Suppose $$G$$ is a finite group and $$k$$ is a splitting field for $$G$$. Then, for any irreducible linear representation $$\rho:G \to GL(n,k)$$, and any element $$g$$ in the center of $$G$$, the image $$\rho(g)$$ is a scalar matrix.

Instead of requiring $$k$$ to be a splitting field, we can require only that $$k$$ have characteristic not dividing the order of $$G$$ and the representation $$\rho$$ be absolutely irreducible.

Related facts

 * Irreducible representation over splitting field surjects to matrix ring

Related facts about character values
Note that in characteristic zero (and with a little extra work, in other characteristics) this fact implies that a central element has a nonzero character value for all irreducible characters. For non-central elements, it may or may not be true that every irreducible character has a nonzero value. Some related results:


 * Conjugacy class of more than average size has character value zero for some irreducible character
 * Irreducible character of degree greater than one takes value zero on some conjugacy class
 * Zero-or-scalar lemma states that if the degree of an irreducible representation and the size of a conjugacy class are relatively prime, then either the character value is zero or the conjugacy class maps to scalar matrices.

Facts used

 * 1) uses::Schur's lemma