All finite groups that have exactly 3 conjugacy classes .

Question

This question was asked in my abstract algebra quiz .

Determine all finite groups which have exactly 3 conjugacy classes .

I solved that $\mathbb{Z}/3$ is only group with conjugacy class 3.

While checking my argument by google search here :https://www.cefns.nau.edu/~falk/old_classes/511/extras/threeclass.html

I found that $S_{3}$ also has 3 conjugacy classes . But I have a question in the argument of the link .

Question :How does in the explanation given in the link author wrote :"m divides 1 + n and n divides 1 + m" in the 3rd line of the argument . I know the result"since the size of a conjugacy class equals the index of the centralizer of one of its elements)" which is given before the deduction but I dont know how to use it to deduce the result.

Can anyone please tell how to deduce the result ?

Answer 1

$|G|-m=1+n$. Since $m$ divides the order of the group and itself, it has to divide $1+n$. The same works for $n$.