# Central implies image under every irreducible representation is scalar

From Groupprops

## Contents

## Statement

Suppose is a finite group and is a splitting field for . Then, for any irreducible linear representation , and any element in the center of , the image is a scalar matrix.

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

### 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

## 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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