State the row orthogonality relations. Prove that if $\chi$ is an irreducible character of the finite group $G$, then $\chi(1)$ divides the order of $G$.

Stating clearly any additional results you use, deduce the following statements:

(i) Groups of order $p^{2}$, where $p$ is prime, are abelian.

(ii) If $G$ is a group of order $2 p$, where $p$ is prime, then either the degrees of the irreducible characters of $G$ are all 1 , or they are

$$1,1,2, \ldots, 2(\text { with }(p-1) / 2 \text { of degree } 2)$$

(iii) No simple group has an irreducible character of degree 2 .

(iv) Let $p$ and $q$ be prime numbers with $p>q$, and let $G$ be a non-abelian group of order $p q$. Then $q$ divides $p-1$ and $G$ has $q+((p-1) / q)$ conjugacy classes.