Linear Group with Nontrivial p-Core is Reducible

Proposition

Let $G⩽\mathrm{GL}(n,p)$ be a linear group. If $O_p(G)\ne 1$, then $G$ is reducible.

\begin{proof} Let $N:=O_p(G)$ be the [[Group Core|$p$-core]] of $G$, that is, the largest normal $p$-subgroup of $G$. Then $X:=V:G$ has a normal subgroup $Y:=V:N$. Since $Y$ is a $p$-group, the center $Z(Y)$ is non-trivial and it is easy to verify that $Z(Y)⊲V$. Thus, $Z(Y)$ is a non-trivial subspace of ${\mathbb{F}}_p^n$. For any $g\in G$, $n\in N$ and $z\in Z(Y)$, there is $(z^g{)}^n=z^{gn{g}^{-1}g}=z^g$ and so $z^g\in Z(Y)$. Consequently, $Z(Y)$ is fixed by $G$ and so $G$ is reducible. \end{proof}

Remark. See also Irreducible Representation of p-Group.