Representation Theory – Representations of Cyclic Group of Order p

representation-theory

Let $p$ be a prime. My eventual goal is to prove that the only irreducible representation of a $p$-group over a field of characteristic $p$ is the trivial representation.

At the moment, I'm trying to prove a simpler claim: suppose $G$ is a (cyclic) group of order $p$, let $K$ be a field of characteristic $p$, and let $V$ be any representation of $G$ over $K$; if the dimension of $V$ (as a vector space over $K$) is greater than $1$, then $V$ must be a reducible representation.

Does anyone have any hints or suggestions on how to proceed? For example, where should I use the fact that $K$ has characteristic equal to $|G|$ (actually, I'm not even sure whether or not the above holds for arbitrary $K$) or the fact that $G$ is cyclic?

Best Answer

Here are some hints:

  1. How many elements does $V$ have? (I am assuming without loss of generality that it is finite-dimensional.)
  2. What do you know about the sizes of orbits of the vectors in $V$ under the action of $G$?

If you can answer these questions, you will immediately be able to prove that the only irreducible representation is the trivial one.

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