so the question is:
(ii) Each irreducible representation of $G_1\times G_2$ is isomrophic to a representation $\rho^1\otimes \rho^2$, where $\rho^i$ is an irreducible representation of $G_i$ ($i=1,2$).
And the proof given is:
I have several questions on this. At first it is not really clear to me why it suffices to show that any class function which is orthogonal to those characters is zero. I don't even get the connection at all...
Furthermore the notation with the $*$ confuses me a lot. I've learned this notation in Linear Algebra as being the adjugate matrix, however in this equation we do not really work with matrices in a direct way, so I figured the author uses this notation for something else. However I've not been able to find any definition for this notation in the book Linear Representations of finite Groups. Can anybody figure out what this $*$ means? Because I already don't understand the whole proof, I sadly won't be able to figure this out by myself. Maybe some of you have seen such a notation and know what it might mean.
I am very thankful for your help, and I am sorry, if my question is not really well written.
Recall the irreducible characters forms a basis of the $\mathbb{C}$-vector space of class functions, so to prove you have all the irreducible characters, it suffices to show there are no class functions not represented by linear combinations of your set of irreducible characters. Since we have an inner product, we just need to show the orthogonal complement to our set of characters is trivial.
And for your second question, $z^*$ is a common notation for the complex conjugate of $z\in\mathbb{C}$.