# Size of conjugacy class need not divide exponent

From Groupprops

## Contents

## Statement

It is possible to have a finite group and a conjugacy class in such that the size of does not divide the exponent of .

This is a non-constraint on the Conjugacy class size statistics of a finite group (?).

## Related facts

### Opposite facts

- Size of conjugacy class divides index of center
- Size of conjugacy class is bounded by order of derived subgroup

### Similar facts

- 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

## Proof

### Example of the alternating group

`Further information: alternating group:A4, element structure of alternating group:A4`

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.