I'm studying off an old qualifying exam, and one of the questions is as follows: If $G$ is a finite group, and $\{ \phi_{i} \}_{i \in I}$ are the irreducible representations of $G$ (I presume over some fixed field, say, $\mathbb{C}$, though the question as written doesn't specify), then $\bigcap_{i} \ker(\phi_i) = \{ e\}$. It gives the hint to consider the regular representation of $G$. The way I read this is "if $h \in G \setminus \{e\}$, then there exists an irreducible representation $\phi : G \to \mathbf{GL}(V)$ of $G$ such that $\phi(h) \neq \operatorname{id}_{V}$". The trouble here is that I don't know how to construct such a representation in such generality. I can generally construct a $\mathbb{C}$-subspace of $\mathbb{C}[G]$ such that $h$ fixes the vector space and isn't the identity, e.g. $W = \operatorname{span}_{\mathbb{C}} \left\{ \sum_{k = 0}^{|h| - 1} e^{2 \pi i k / |h|} h \right\}$, but in general this isn't a $G$-invariant subspace.
I don't know where to go with this one, and would appreciate any advice. Thanks!
I assume that we are working over a field $K$ with $\operatorname{char}(K) \nmid |G|$, so that $K[G]$ decomposes into irreducible representations by Maschke’s Theorem.
An element $g \in G$ is contained in $\bigcap_{i \in I} \ker(\phi_i)$ if and only if it acts trivially on every irreducible representation of $G$. Since $K[G]$ decomposes into irreducible representations it then follows that $g$ acts trivially on $K[G]$. But then $g = g \cdot 1_{K[G]} = 1_{K[G]} = 1_G$.