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of a finite group , the order of the subgroup divides the order of the group; that is, is a divisor of . In particular, the order of any element is a divisor of .

Example

The symmetric group S₃ has the following multiplication table.
:

This group has six elements, so . By definition, the order of the identity, , is one, since . Each of , , and squares to , so these group elements have order

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