Conjugacy class size statistics of a finite group
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Contents
Definition
Let be a finite group. The conjugacy class size statistics of
is a function
that outputs, for each
, the number of conjugacy classes of
of size
. Note that since size of conjugacy class divides order of group, the function is nonzero only on (some) divisors of the order of
.
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 .
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:
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.