# Group Theory: Proving Abelian of Order 4 or Less

• tuggler
In summary, to show that any group of order 4 or less is abelian, we can use the corollary of Lagrange to determine that the elements of the group must have orders of 2 or 4. There are two possibilities: all elements have order 2, or at least one has order 4. This is because the order of an element must divide the order of the group. The element of order 1 is reserved for the identity element.

## Homework Statement

Show that any group of order 4 or less is abelian

2. The attempt at a solution

I came across this hint.

Since its of order 4 we have {e,a,b,c}, where e = identity.

The elements a, b, c must have order 2 or 4. There are two possibilities.

1. a, b, c all have order 2.
2. At least one of them, say a, has order 4.

Why does the elements a, b, c have order 2 or 4? And why does at least one of them have order 4?

Order is the exponent you have to raise them to obtain the identity?
If the group is of order 4 you can suppose that you have at least one element with order of 3 and arrive a contradiction.

SqueeSpleen said:
Order is the exponent you have to raise them to obtain the identity?

Are you asking me a question or is that the answer to my question? lol

tuggler said:
Why does the elements a, b, c have order 2 or 4? And why does at least one of them have order 4?
The order of elements of the group have to divide the order of the group (Corollary of Lagrange) This does not tell us for sure that a group of order 4 has elements of order 2, only that 2 and 4 are the only possible orders. The element of order 1 is reserved as the identity element.

tuggler said:
Are you asking me a question or is that the answer to my question? lol

Sorry, I didn't meant to confuse you, I was asking if the order of an element was the minimum exponent you need to raise it to obtain the identity. In other words, I was assuming that was the property "order" in my answer.

Thank you guys!