# Size of conjugacy class need not divide index of abelian normal subgroup

From Groupprops

## Statement

It is possible to have a finite group , a conjugacy class of , and an abelian normal subgroup of such that the size of does not divide the index .

This puts a non-constraint on the sizes of conjugacy classes.

## Proof

### Example of symmetric group of degree three

`Further information: symmetric group:S3, element structure of symmetric group:S3, subgroup structure of symmetric group:S3`

In symmetric group:S3, a group of order six, there is a subgroup A3 in S3 that is abelian, normal, and has index two.