Can anybody help me with these questions as I am completely lost and haven't done any character theory or rep theory for a long time.
(1) Let G be a group of odd order (so |G| ≡ 1 mod 2) and (conjugacy) class number h. Prove that |G| ≡ h mod 16.
For example, has two irreducible characters, one of which is the trivial character. The reason why irreducible characters come in pairs is that the finite group G is of odd order by hypothesis. As my previous post showed, if G is of odd order, the trivial character is the only irreducible character that is real valued, and other irreducible characters are paired.