Characters of linear representations

Let $\Phi , \Psi$ be irreducible complex representations of a finite group $G$. Obtain the generalization of Theorem 2

Here $h$ is an arbitrary element of the group $G ; \delta_{\Phi , \Psi }=1$ or $\delta_{\Phi , \Psi }=0$ according as $\Phi$ and $\Psi$ are equivalent or inequivalent.

Apply the character-based irreducibility criterion to the representation $\Phi^{(3)}$ of the group $S_{3}$ from Example 1, item $1, \S 2$.

Prove, using Schur's lemma, that all irreducible representations over $\mathbb {C}$ of an abelian group $G$ are one-dimensional.

If a group $G$ has an automorphism $\tau$, then with every linear representation $(\Phi , V)$ of this group one associates another representation $\left(\Phi^{\tau }, V\right)$, defined by the rule $\Phi^{\tau }(g)=\Phi (\tau (g))$. Verify that this is indeed so, and show that the irreducibility of $\Phi$ implies the irreducibility of $\Phi^{\tau }$. As a rule, $\Phi^{\tau } \approx \Phi$, but there are cases when a new representation results. What should one expect in the case of an inner automorphism?

Let $G=A_{5}$ and let $\Phi$ be the representation considered in Example 2. The map $\tau : \pi \mapsto (12) \pi (12)^{-1}$ is an (outer) automorphism of the group $A_{5}$, permuting the classes with representatives (12345) and (12354). The sets of values of the characters $\chi$ and $\chi^{\tau }$ are obtained from one another by interchanging $(1+\sqrt{5}) / 2$ and $(1-\sqrt{5}) / 2$. Show that the characters $\chi^{\tau }$ and $\chi$ are inequivalent.

Let $\Phi : G \longrightarrow U(n), \Psi : G \longrightarrow \mathrm{U}(n)$ be equivalent unitary irreducible representations of a finite group $G$. Prove that there exists a unitary matrix $C$ for which $C \Phi_{g} C^{-1}=\Psi_{g} \quad \forall g \in G$.

Prove that the center $Z(G)$ of a finite group $G$ possessing a faithful irreducible representation over $\mathbb {C}$ is always trivial or cyclic.

Let $g \mapsto \Phi_{g}, g \mapsto \Psi_{g}$ be two matrix complex representations of a finite group $G$. Suppose that for each $g \in G$ there is a nonsingular matrix $C_{g}$ such that $C_{g} \Phi_{g} C_{g}^{-1}=\Psi_{g}$. Prove that there exists a nonsingular matrix $C$, independent of $g$, for which $C \Phi_{g} C^{-1}=\Psi_{g}$.