Order of Groups | Order of an element in a Group

The order of a group and its elements are very crucial in group theory (Abstract Algebra). One can study groups by analyzing the orders of the group and their elements. In this article, we will learn about the order of groups.

Order of a group

The order of a group G is the cardinality of that group. In other words, the order of a group G is the number of its elements.

Notation: The order of a group G is denoted by |G| or $\circ$(G).

The following are a few examples on orders of groups.

  • The group (Zn, +) is a group of order n.
  • The symmetric group (S3, 0) has order 6.
  • (Z, +) is a group of infinite order.

Types of Groups

Depending upon the order of groups, we can classify the groups as follows:

  1. Finite Group: If a group contains a finite number of elements, then it is a finite group. For example, the symmetric group Sn is a finite group of order n!.
  2. Infinite group: If a group does not have a finite number of elements, then it is an infinite group. For example, the additive group of integers is not a finite group; it is an infinite group.

Order of an element

Order of elements in a group G. The order of an element a ∈ (G, 0) is the smallest positive integer n such that an=e, where e is the identity element of G. It is denoted by |a| or $\circ$(a).

If such an n exists, then we say that the element a is of finite order. Otherwise, a is said to be an element of infinite order.

Order of an Element Formula

Let a be an element of order n in a group (G, 0), that is, an=e. Then the order of ak is given by the formula:

0(ak) = $\dfrac{n}{\text{gcd}(k, n)}$.

We will now understand this formula with an example.

Example: Let a be an element of order 20 in a group (G, 0). Find the order of a5.

Solution:

We are given 0(a)=20, that is, a20=e.

In the above formula, we have n=20 and k=5. So the order of a5 is

0(a5) = $\dfrac{20}{\text{gcd}(5, 20)}$ = 20/5 = 4.

Properties of Order of Elements in a Group

The order of an element of a group satisfies the below properties:

  • The order of the identity element in a group is 1. No other element has order 1.
  • Both an element and its inverse of a group have the same order. In other words, 0(a)= 0(a-1) for all elements a in G.
  • Each element of a finite group has finite order and it divides the order of the group (Lagrange’s Theorem). Thus no element exists in a finite group whose order exceeds the order of the group.
  • If 0(a) = k and an=e, then k is a divisor of n.
  • Suppose 0(a) = k. Then 0(an) = k for every integer n coprime to k.
  • If 0(a) is infinite, then 0(an) is also infinite for every integer n.
  • Both a and g-1ag have the same order for a, g ∈ G. These two elements are called conjugate elements of each other.
  • 0(ab) = 0(ba) for all a, b ∈ G, that is, both ab and ba have the same order. Because ab and ba are conjugates of each other as ab=a(ba)a-1

FAQs on Order of Groups and its Elements

Q1: What is the order of the identity element?

Answer: In a group, the order of the identity element is 1.

Q2: What is the order of Z12?

Answer: As Z12 contains 12 elements, the order of Z12 is 12.