Central implies image under every irreducible representation is scalar
Instead of requiring to be a splitting field, we can require only that have characteristic not dividing the order of and the representation be absolutely irreducible.
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.