Size of conjugacy class need not divide exponent

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Statement

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

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

Related facts

Opposite facts

Similar facts

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.