Order of element divides order of group

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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 G be a finite group and g \in G be an element. Let m be the order of the element g: the smallest positive integer m such that g^m is the identity element. Then, m divides the order of G. In particular, we have, for any g \in G, that:

g^{|G|} = e.


Facts used

  1. Lagrange's theorem

Related facts

Applications

Converse

Proof

The proof follows from Lagrange's theorem, along with the observation that the order of the element g equals the order of the cyclic subgroup gneerated by g, which is therefore a subgroup of G.