Thread: Determining if a group

Hi there,
I'm doing some revision for my exam and I cannot see how this one particular solution has come about. The question is:

Is a semigroup, or group in the following?
(a) \ ,

I have solved a lot of questions like this, so I know its a group because i have been able to show that it has an identity and inverse, and i have the solutions to this problem so i have seen the associative property proved, but that is where I am getting stuck

The solution I have says (I'll number each line)....
(1)
(2)
(3)
(4)
(5)
(6)

Now normally what I like to do when looking to see if a particular operation is associative is do steps (1), (2), (3), and then do (6), (5), (4), (3) and see if i get the same result for (3) both times. However in this case I cannot see how (2) was obtained from (1). Initially I thought that step 2 would be:
. I cannot see why (2) is . Can anyone please explain why this is the case?
Thanks in advance to those who can help.
Cheers,
Maccaman

To easy confision let us write multiplication as . So when going from line 1 to line 2 we have that . , and the result follows.

The and in the definition of multiplication and associativity are just arbitrary elements from the group.

I see. I must have been tired or something to miss that. Thanks very much Swlabr. You