I have a question about representation theory.
Let $G$ be a finite group with a unique minimal normal subgroup, let $\mathbb{F}$ be a field such that its characteristic is coprime to $|G|$. I need to prove that $G$ admits a faithful irreducible representation. The irreducibility is a consequence of Maschke's Theorem, but how could I prove the faithfulness?
Thank you very much!
Best Answer
Consider a regular representation of $G$ i.e. $G$ acts on the left on the group $\mathbb{F}$-algebra $F[\mathbb{G}]$. This is a faithful finite dimensional representation. Therefore, the class of finite dimensional faithful representations of $G$ is nonempty. Pick now a faithful and finite dimensional representation $\rho:G\rightarrow \mathrm{GL}(V)$. Decompose $V$ as a direct sum of irreducible representations
$$V = V_1\oplus V_2\oplus ...\oplus V_k$$
of $G$ (this requires $|G|$ coprime to characteristic of $\mathbb{F}$). For every $i=1,2,...,k$ consider $N_i\subseteq G$ the set of $g\in G$ such that $\rho(g)$ acts trivially on $V_i$. Then $N_i$ is a normal subgroup of $G$ (it is a kernel of the induced representation $\rho_{\mid V_i}$). Moreover, since $\rho$ is faithful, we derive that $$\bigcap_{i=1}^kN_i = \{1\}$$ Let $N$ be a minimal (with respect to inclusion) nontrivial normal subgroup of $G$. If each $N_i$ is nontrivial, then $N\subseteq N_i$ and hence $$N\subseteq \bigcap_{i=1}^kN_i$$ This is a contradiction and hence there exists $i_0\in \{1,2,...,k\}$ such that $N_{i_0}= \{1\}$. Therefore, $\rho_{\mid V_{i_0}}$ is irreducible and faithful.