A group of order 4 is always abelian or commutative. That is, ab = ba for all a, b in G where |G| =4. In this article, we will prove that each group of order 4 is abelian.

**What are Abelian Groups?**

A pair (G, o) is called an abelian group if G is closed under the binary operation o, o is associative, the identity element exists, every element of G has an inverse in G, and ab = ba for all a, b ∈ G.

Order of Groups: The number of elements of the group G is called the order of G. It is denoted by |G|.

Table of Contents

## Prove that Every Group of Order 4 is Abelian

**Proof:**

Let G be a group of order 4.

So |G| = 4.

As the order of an element of a group divides the order of the group, we conclude that the order of elements of G can be 1, 2, or 4.

Case 1 : |

The group G contains an element of order 4.

As G contains an element of order 4 = |G|, so by the properties of cyclic groups, we conclude that the group G must be cyclic.

As every cyclic group is abelian, G must be abelian. In this case, G is isomorphic to the group ℤ/4ℤ.

Case 2 : |

The group G does not contain an element of order 4.

So each non-identity element has order 2. We write

G = {e, a, b, c}.

As a, b ∈ G, we have ab, ba∈ G.

We claim that ab = c.

Using the cancellation law, we have the following.

ab = a | ab = ae ⇒ b=e, not possible. |

ab = b | a=e, which is also not possible. |

ab = e | ab =a^{2} as a has order 2 ⇒ b = a, which is a contradiction. |

So it follows that ab = c.

In a similar way, we can show that ba = c. Hence, ab = ba. This shows that every non-identity element commutes each other. This further implies that G is abelian.

In this case, G is isomorphic to the group ℤ/2ℤ × ℤ/2ℤ.

**Related Topics:**

- Group Theory, Definition, Examples, Theorems
- Abelian Group: Definition, Examples, Properties
- Orbit Stabilizer Theorem
- Two Cyclic Groups of Same Order are Isomorphic
- Quotient Group
- Normal Subgroup
- Simple Group

## FAQs

**Q1: Is a group of order 4 abelian/commutative?**

Answer: Yes, a group of order 4 is always abelian or commutative. This is because if a group has order 4, then it must be isomorphic to ℤ/4ℤ or ℤ/2ℤ × ℤ/2ℤ. In either case, the group is abelain.

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.