Central implies image under every irreducible representation is scalar

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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

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:

Facts used

  1. Schur's lemma

Proof

Proof over an algebraically closed field

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Why the proof continues to work over a splitting field

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