Irreducible representation of group of prime power order in characteristic equal to underlying prime is trivial

Statement
Suppose $$G$$ is a group of prime power order where the underlying prime is $$p$$. Suppose $$K$$ is a field of characteristic $$p$$ (such as the prime field $$\mathbb{F}_p$$ or any extension thereof). Then, any irreducible linear representation of $$G$$ over $$K$$ is the trivial representation.

Related facts
Here are some facts proved with the same tools:


 * Prime power order implies not centerless
 * Prime power order implies center is normality-large

Facts used

 * 1) uses::Class equation of a group action