Order of element divides order of group
This article states a result of the form that one natural number divides another. Specifically, the (order of an element) of a/an/the (element of a group) divides the (order of a group) of a/an/the (group).
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Let be a finite group and be an element. Let be the order of the element : the smallest positive integer such that is the identity element. Then, divides the order of . In particular, we have, for any , that:
- Cauchy's theorem: This states that for every prime dividing the order of a group, there is an element whose order equals that prime number.
- Sylow's theorem: This asserts the existence of -Sylow subgroups for every prime dividing the order of the group.
- Exponent of a finite group has precisely the same prime factors as order: This is a consequence of Cauchy's theorem.
The proof follows from Lagrange's theorem, along with the observation that the order of the element equals the order of the cyclic subgroup gneerated by , which is therefore a subgroup of .