Problem:Determine if the set with the binary operation forms a group.

Set S = All real numbers except -1.

$\displaystyle a*b = ab+a+b$

****************

I know I need to check three things: associativity, if there is an identity, and if there is an inverse for every element in S.

I know how to do associativity since it's straightforward. For the identity, I just guess and checked and found that $\displaystyle e=0$ works. (Is there any way to find an identity without guessing/checking?)

For the inverse, I am having difficulty.

I know that this must be satisfied to have an inverse:

$\displaystyle a*a' = a'*a = e$

Since I know the identity element is 0,

$\displaystyle a*a' = a'*a = 0$

Expanding this out with the definition of the binary operation,

$\displaystyle aa'+a+a' = a'a+a'+a = 0$

I said that an inversedoes not exist, because from the last equation, it is impossible to get $\displaystyle aa'+a+a'$ equal to 0 for all a. And therefore, this is not a group.

Am I correct?