Determine if the set with the binary operation forms a group.
Set S = All real numbers except -1.
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 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:
Since I know the identity element is 0,
Expanding this out with the definition of the binary operation,
I said that an inverse does not exist, because from the last equation, it is impossible to get equal to 0 for all a. And therefore, this is not a group.
Am I correct?