Conjugacy class size statistics of a finite group

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Definition

Let G be a finite group. The conjugacy class size statistics of G is a function f:\mathbb{N} \to \mathbb{N}_0 that outputs, for each d, the number of conjugacy classes of G of size d. Note that since size of conjugacy class divides order of group, the function is nonzero only on (some) divisors of the order of G.

The conjugacy class size statistics carry more information than the conjugacy class size set of a finite group, which is simply the set of sizes of the conjugacy classes in G.

Relation with other statistics

Stronger statistics

Facts

Facts about conjugacy class sizes

Divisibility facts:

Bounding facts:

Non-divisibility/non-bounding facts:

Relation with degrees of irreducible representations

The number of conjugacy classes is an important measure in relating conjugacy classes to irreducible representations and to the degrees of irreducible representations.