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 $\displaystyle (G, *) $ a semigroup, or group in the following?

(a) $\displaystyle G = \mathbb{R} \ $\$\displaystyle \{\frac{1}{2}\}$, $\displaystyle a * b = a - 2ab +b $

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) $\displaystyle (a * b) * c = (a - 2ab + b) * c $

(2) $\displaystyle \ \ \ \ \ \ \ = (a - 2ab + b) - 2(a - 2ab + b)c + c $

(3) $\displaystyle \ \ \ \ \ \ \ = a + b + c - 2ab - 2ac - 2bc + 4abc $

(4) $\displaystyle \ \ \ \ \ \ \ = a - 2a(b - 2bc + c) + (b - 2bc + c) $

(5) $\displaystyle \ \ \ \ \ \ \ = a * (b - 2bc + c) $

(6) $\displaystyle \ \ \ \ \ \ \ = a * (b * c) $

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:

$\displaystyle = (a - 2ab + b) + c - 2c $. I cannot see why (2) is $\displaystyle \ \ \ \ \ \ \ = (a - 2ab + b) - 2(a - 2ab + b)c + c $. Can anyone please explain why this is the case?

Thanks in advance to those who can help.

Cheers,

Maccaman