Prove that the following set is a group Prove that that the following is or is not a group.
(a) The set S = $\mathbb{R}$ \ {0} with operation defined by a * b = 2ab for all a and b in S. (On the right side of the equation, the operations are the usual addition and multiplication in $\mathbb{R}$.)
(b) The set S = $\mathbb{R}$ with operation defined by a * b = 2(a + b) for all a and b in S.
I do not have the slightest clue how to begin. I know that to be a group the binary operation must be associative, the group must have an identity, and every element in the group must have an inverse. 
The trouble I am having is identifying what the identity is in a) and starting the proof. I am also confused about part a) in that I do not know what equation is being referred to in the parenthesis. Please help?

The identity in (a) will be $1/2$: if $a * b = a$, then $2ab = a$, $a(2b-1) = 0$ and therefore we are forced to set $b = 1/2$ (because $*$ is commutative).
In (b) notice that the operation is simply the usual addition multiplied by two and $(\mathbb R, +)$ is a group.
