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

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Statement

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

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.