# Representation pullbackability theorem

This fact is related to: representation theory
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## Statement

Let $G$ be a finite group and $k$ a finite field whose characteristic does not divide the order of $G$. Let $A$ be a local ring with residue field $k$. Then, any representation of $G$ over $k$ pulls back to a representation of $G$ over $A$.

In other words, given any map:

$\rho:G \to GL_n(k)$

there exists a map:

$\tilde{\rho}:G \to GL_n(A)$

such that $\tilde{\rho}$, composed with the residue map modulo the maximal ideal, gives $\rho$.

## Proof

$k$ has order $q$ where $q=p^l$, $p$ not dividing the order of $G$. Let $\pi$ be the map from $GL_n(A)$ to $Gl_n(k)$, that involves taking each matrix entry modulo the maximal ideal.

The kernel of $\pi$ is, as can easily be checked, the subgroup comprising matrices which are congruent entry-wise to the identity matrix. The cardinality of this kernel is thus $p^{ln^2}$.

Then, consider $\pi^{-1}(\rho(G))$. The map:

$\pi: \pi^{-1}(\rho(G)) \to \rho(G)$

has, as kernel, a $p$-group. Since $p$ does not divide the order of $G$, it does not divide the order of $\rho(G)$, os the kernel is a normal Sylow subgroup. We can thus apply Schur-Zassenhaus theorem to find a section for this map. Composing this section with $\rho$ gives us the pulled back representation $\tilde{\rho}$.