Let $G$ be a finite group and $F$ a field with $\text{char}(F) \nmid \# G .$ We say that a conjugacy class $C$ of $G$ is even if $g \in C$ implies $g^{-1} \in C$.
Show that if every conjugacy class of $C$ is even, then every finite-dimensional representation $V$ of $G$ over $F$ satisfies $V\simeq V^*$.
Could you please help me with the above problem?
Now assume that $F$ splits $G$. Show that the number of even conjugacy classes of $G$ equals the number of isomorphism classes of irreducible representations $V$ of $G$ over $F$ satisfying $V \simeq V^*$.
We have "the number of conjugacy classes of $G$ equals the number of isomorphism classes of irreducible representations $V$ of $G$ over $F$". Is the second conclusion trivial?
By the way, why it requires "$F$ splits $G$"?
Thanks.