Abelian Group: Definition, Properties, Examples

Abelian groups are special types of groups in which commutativity holds. In other words, the binary operation on such groups is commutative. Abelian groups (also called commutative groups) are named after mathematician Niels Henrik Abel. In this article, we will discuss abelian groups with their properties.

What is an Abelian Group?

Definition: A group G is called an abelian (or commutative) group if the group operation o is commutative, that is, a o b = b o a ∀ a, b ∈ G holds.

For example, the set ℤ of integers is an abelian group under addition and it is denoted by (ℤ, +)

Abelian groups are also known as commutative groups. More specifically, if G is a non-empty set and o is a binary operation on G, then the algebraic structure (G, o) is called an abelian group if the following holds:

• The set G is closed under the operation o.
• The binary operation o is associative on G.
• G contains an identity element.
• Every element of G has an inverse in G.
• The operation o is commutative on G, that is, aob = boa ∀ a,b ∈ G.

Examples of Abelian Groups

The following are a few examples of abelian groups.

• (ℤ, +) is an abelian group as a+b=b+a for all a, b ∈ ℤ.
• The set ℚ of all rational numbers is a commutative group under the operation +, that is, (ℚ, +) is an abelian group.
• The group of n-th roots of unity under multiplication is an abelian (commutative) group.
• (ℤ_n, +) is an abelian group.

Non-Abelian Group

If a group G is not abelian, then G is called a non-abelian group. Non-abelian groups are also known as non-commutative groups. Examples of non-abelian groups are given below.

Non-examples of abelian groups: The symmetric group S_n; in particular S₃, is not abelian. They are non-abelian or non-commutative groups.

Properties of Abelian Groups

• Every subgroup of an abelian group is abelian.
• Any cyclic group is abelian.
• Every factor (or quotient) group of a group is abelian.
• The direct product of abelian groups is also abelian.
• The center of an abelian group is abelian.
• The commutator of two elements x, y of a group G is defined by x^-1y^-1xy. Thus if G is an abelian group, then the commutator of any two elements of G is the identity element of G. Hence the derived subgroup of an abelian group G is trivial.
• Every group of prime order is cyclic, and hence abelian.
• If G is a group of order p² where p is a prime number, then either $G \cong Z/p^2Z$ or $G \cong Z/pZ \oplus Z/pZ$. Hence a group of order p² is abelian.
• Let p and q be two primes such that p>q and q $\nmid (p-1)$. If G is a group of order pq, then $G \cong Z/pqZ$, and hence abelian.

Read These:

Basics of Group Theory

Group Homomorphism

Kernel of a Group Homomorphism

First Isomorphism Theorem of Groups

Question 1: Show that (ℤ, +) is an abelian group.

Solution:

(1) For any two integers a and b, the sum a+b is an integer. Thus ℤ is closed under +.

(2) We know that a+(b+c) = (a+b)+c for any a, b, c ∈ ℤ. Thus the operation + is associative on ℤ.

(3) a+0=a for all a ∈ ℤ. So 0 is an identity element in ℤ.

(4) As a+(-a)=0 for all a ∈ ℤ, we say that -a is the inverse element of a.

(5) a+b = b+a for all a,b ∈ ℤ. So + is commutative.

Thus (ℤ, +) is a group that is commutative. Hence (ℤ, +) is an abelian group.

NOTE: In a similar way, one can show that (Q, +), (R, +) all are abelian groups.

Also Read

Second Isomorphism Theorem | Third Isomorphism Theorem

Normal Subgroup | Left and Right Cosets

Center of a Group | Normaliser in a Group

Solved Problems

Question 1: Is every subgroup of an abelian group cyclic?

Answer:

No, not every subgroup of an abelian group is necessarily cyclic. For example, the group ℤ/2 × ℤ/2 × ℤ/2 is an abelian group, but its subgroup ℤ/2 × ℤ/2 is not cyclic as it does not contain an element of order 4 = |ℤ/2 × ℤ/2|.

FAQs on Abelian Groups