Order of element divides order of group
From Groupprops
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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Statement
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:
.
Facts used
Related facts
Applications
Converse
- 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.
Proof
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
.