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.
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.
I assume you're acquainted with the basics of rep. theory. One can prove that for non-trivial characters of the characters of come in pairs of equal degree, say say with degress . So, . But, one has that if is the number of conjugacy classes then . Now, since is odd one has that each is odd, say equal to . Then, plugging this into the equation will give the desired result.
It is not necessarily true that irreducible characters of finite group G come in pairs like
.
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.