The order of an element divides the order of a finite group. If G is a finite group and a ∈ G, then o(a) divides |G|. Let us now recall the definition of the order of an element in a group.

Let a ∈ G. The order of a in G is the smallest positive integer n such that a^{n} = e, the identity element in G. The order of a is denoted by o(a). In this article, we will show that n divides |G|.

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## Proof

Let a ∈ G be an element of order n in a finite group G.

Consider the subgroup generated by a, and denote it by H.

So H = <a>.

Then by definition of cyclic groups, o(a) = |H|.

As H is a subgroup of a finite group G, so by Lagrange’s theorem of finite groups, we deduce that |H| divides |G|.

Noting o(a) = |H|, we obtain that o(a) divides |G|.

So the order of an element in a finite group divides the order of that group.

## Solved Problems

**Question 1:** Is there any element of order 4 in Z_{6}?

**Answer:**

The group Z_{6} has order 6.

As 4 does not divide 6, by the above fact we conclude that there does not exist any element of order 4 in Z_{6}.

**More Topics:** An Introduction to Group Theory

Abelian Group | Quotient Group

Normal Subgroup | Simple Group

Prove that a group of prime order is cyclic

Every Subgroup of a Cyclic Group is Cyclic: Proof

## FAQs

**Q1: What is the order of an element in a group?**

Answer: The order of an element in a group is the least positive integer n such that the n-th power of that element is the identity of the group. For example, the identity element has order 1.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.