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

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Statement

In finite terms

Let d be any positive integer. Then, the following hold:

  1. There are only finitely many finite groups with exactly d 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 number of conjugacy classes of G by n(G). Then:

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

Facts used

  1. Size of conjugacy class equals index of centralizer (we use it to derive a primitive version of the class equation of a group)
  2. 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_is. 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 Gs 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_is 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_is that equal a_1 also divides a_1 (because this number is the order of the center).
  3. If the number of a_is that equal a_1 is \alpha, then all the a_is 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.

Number of conjugacy classes d Egyptian fraction representation of unity Groups where this representation is realized
1 1 trivial group
2 1/2 + 1/2 cyclic group:Z2
3 1/3 + 1/3 + 1/3 cyclic group:Z3
3 1/4 + 1/4 + 1/2 not realized for any group -- condition (3) violated.
3 1/6 + 1/3 + 1/2 symmetric group:S3
4 1/4 + 1/4 + 1/4 + 1/4 cyclic group:Z4, Klein four-group
4 1/6 + 1/3 + 1/4 + 1/4 not realized for any group -- condition (1) violated
4 1/6 + 1/6 + 1/3 + 1/3 not realized for any group -- condition (3) violated
4 1/6 + 1/6 + 1/6 + 1/2 not realized for any group -- condition (3) violated
4 1/8 + 1/8 + 1/4 + 1/2 not realized for any group -- condition (3) violated
4 1/10 + 1/5 + 1/5 + 1/2 dihedral group:D10
4 1/12 + 1/4 + 1/3 + 1/3 alternating group:A4
4 1/12 + 1/12 + 1/3 + 1/2 not realized for any group -- condition (3) violated