Size of conjugacy class need not divide exponent
This is a non-constraint on the Conjugacy class size statistics of a finite group (?).
- Size of conjugacy class divides index of center
- Size of conjugacy class is bounded by order of derived subgroup
- Size of conjugacy class need not divide index of abelian normal subgroup
- Size of conjugacy class need not divide order of derived subgroup
Related facts about degrees of irreducible representations
Example of the alternating group
In the alternating group of degree 4, the elements have orders 1,2,3, so the exponent is 6. However, there are two conjugacy classes of size 4, both comprising 3-cycles. 4 does not divide 6.