Conjugacy class size statistics of a finite group

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

• Conjugacy class root statistics of a finite group
• Conjugacy class-cum-order statistics of a finite group

Facts

Facts about conjugacy class sizes

Divisibility facts:

• Size of conjugacy class divides order of group
• Size of conjugacy class divides index of center
• Size of conjugacy class equals index of centralizer

Bounding facts:

• Size of conjugacy class is bounded by order of derived subgroup

Non-divisibility/non-bounding facts:

• Size of conjugacy class need not divide exponent
• Size of conjugacy class need not divide index of abelian normal subgroup
• Size of conjugacy class may be greater than index of abelian normal subgroup

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.

• Number of irreducible representations equals number of conjugacy classes
• Degrees of irreducible representations need not determine conjugacy class size statistics
• Conjugacy class size statistics need not determine degrees of irreducible representations