# Group thoery

1. Sep 19, 2013

### tuggler

1. The problem statement, all variables and given/known data

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?

2. Sep 19, 2013

### SqueeSpleen

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.

3. Sep 19, 2013

### tuggler

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

4. Sep 19, 2013

### CAF123

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.

5. Sep 19, 2013

### SqueeSpleen

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.

6. Sep 19, 2013

### tuggler

Thank you guys!