# Representation of a finite group over a finite field from rational representations

Suppose $$G$$ is a the cyclic group of $$n$$ elements and $$p$$ is a prime not in $$n$$. $$G$$ has an action on the cyclotomic field as a $$\mathbb{Q}$$-vector space $$\mathbb{Q}[X] / \langle \Phi_n(X) \rangle$$ by multiplication by $$X$$. This is an irreducible $$\mathbb{Q}[G]$$-representation. In particular, it acts as automorphisms on $$\mathbb{Z}[X]/\langle \Phi_N(X) \rangle$$. After reduction modulo $$p$$, $$G$$ acts on $$\mathbb{F}_p[X]/\langle \Phi_N(X) \rangle$$. However, after reduction mod $$p$$, the representation is no longer irreducible and splits into isomorphic copies the same irreducible representations. This can be shown by seeing how the polynomial $$\Phi_N(X)$$ splits in the finite field.

One can now consider a finite group $$G$$ and any prime $$p$$ not appearing in $$|G|$$. If we have an irreducible $$\mathbb{Q}[G]$$-representation $$G\rightarrow GL_n(\mathbb{Q})$$, after some conjugation, we can assume $$G \rightarrow GL_n(\mathbb{Z})$$ and then consider the induced $$\mathbb{F}_p$$-representation. Are there any similar results that generalize the above result of cyclic groups and let us know something about the irreducible components?

You basically have to look at the irreducible $$\mathbf C$$-representation containing the $$\mathbf Q$$-representation and play around with classical decomposition theory and Schur indices.