There are finitely many finite groups with bounded number of conjugacy classes

In finite terms
Let $$d$$ be any positive integer. Then, the following hold:


 * 1) There are only finitely many finite groups with exactly $$d$$ fact about::conjugacy classes of elements.
 * 2) There are only finitely many finite groups with at most $$d$$ conjugacy classes of elements.
 * 3) There exists a number $$n_d$$ dependent on $$d$$ such that any finite group of order more than $$n_d$$ has more than $$d$$ conjugacy classes.

In limit terms
Denote, for a finite group $$G$$, the fact about::number of conjugacy classes of $$G$$ by $$n(G)$$. Then:

$$\lim_{|G| \to \infty} n(G) = \infty$$

Facts used

 * 1) uses::Size of conjugacy class equals index of centralizer (we use it to derive a primitive version of the uses::class equation of a group)
 * 2) uses::Number of Egyptian fraction representations of unity with bounded number of fractions is finite: For fixed $$d$$, there are only finitely many solutions to:

$$\sum_{i=1}^d \frac{1}{a_i} = 1$$

where the $$a_i$$ are (not necessarily distinct) positive integers.

Proof
We prove that there is a bound on the order of any finite group $$G$$ with exactly $$d$$ conjugacy classes.

For a finite group $$G$$ of order $$n$$ with $$d$$ conjugacy classes, let $$a_1 \ge a_2 \ge \dots \ge a_d$$ be the sizes of the centralizers of representatives of each of the conjugacy classes, arranged in descending order. By fact (1), the size of the $$i^{th}$$ conjuacy class is $$|G|/a_i$$. Since the group is a union of its conjugacy classes, we get:

$$\frac{|G|}{a_1} + \frac{|G|}{a_2} + \dots + \frac{|G|}{a_d} = |G|$$

Canceling $$|G|$$ from both sides, we get:

$$\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_d} = 1$$

Note that $$a_1 = |G|$$ so we can recover $$|G|$$ from the values of the $$a_i$$s. By fact (2), the number of positive integer solutions $$(a_1,\dots,a_d)$$ to the above system is finite. Taking the maximum of the possible values of $$a_1$$ among all these gives a finite upper bound on the order of $$G$$, and hence a finite limit on the number of possible $$G$$s that have $$d$$ conjugacy classes.

Particular cases
Note that there are a number of additional necessary (but not sufficient) conditions for a decomposition:

$$\sum_{i=1}^d \frac{1}{a_i} = 1$$

to arise from a group, with $$a_1 \ge a_2 \ge \dots a_d$$. These include:


 * 1) All the $$a_i$$s must divide $$a_1$$, since by Lagrange's theorem, any centralizer is a subgroup and hence must divide the order of the group.
 * 2) The number of $$a_i$$s that equal $$a_1$$ also divides $$a_1$$ (because this number is the order of the center).
 * 3) If the number of $$a_i$$s that equal $$a_1$$ is $$\alpha$$, then all the $$a_i$$s are multiples of $$\alpha$$. For a nontrivial group, all of them are strictly bigger than $$\alpha$$.

Below are some examples. The list of Egyptian fraction representations is exhaustive for $$d=1,2,3$$, but there are many other cases for $$d = 4$$ that we have not listed here.