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 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 |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
FAQs
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.