Let be a finite group and be the number of conjugacy classes of It's easy to see that the probability that two elements of chosen randomly, commute is equal to
We define Obviously if and only if is abelian. So measures how "far" the group is from being abelian. The most basic question is to see whether
or not this "distance" can be made as small as we wish. The answer to this question is surprisingly negative:
Problem: Using the class equation, or any way you like, prove that for any finite non-abelian group
Remark: the dihedral group of order has exactly conjugacy classes and thus
Note: There are many suprising and beautiful results, not all easy to prove, about I have already posted an easy one here. I will discuss more results later.