Ok, for this one you only need some very basic knowledge of group theory:
Let be a finite non-abelian group and Let be the number of conjugacy classes of . Prove that with equality if and only if
Let be the conjugacy classes of . Assume WLOG that the first are non-central.
Suppose Then Thus We will reach a contradiction for each case.
The case is impossible since this would imply were abelian. So assume .
Let with . Let with . Consider
Since and are both noncentral and has order two we know that
Hence, we have shown that for all But this implies which is a contradiction.
I don't have time to complete the last case. But it is fairly similar. Again use the fact that we know the group structure of and find the similar contradiction. Then use the class equation to get the inequality for . I have not proven the case of equality but I'll leave that for someone else.
combine this result with the class equation to prove the first part. the proof of the second part of the problem comes from the proof of the first part.