Group of Order 4 is Abelian: Proof

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|.

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 =a2 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

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