Identity element of a group So I already proved Closure and Associativity, now I'm trying to find the identity element of this operation defined as:
$$
a * b = a + b - ab
$$
But my identity element gets cancelled...
(The set defined in this exercise is the real numbers.)

 A: Claim: The identity element in $(\mathbb{R},*)$ is the real number zero.
Proof: For any $x\in \mathbb{R}$, $x*0=x+0-x\times 0=x$. Since the identity element in a group is unique, zero is the identity element.
Following your way, suppose the identity is $e$, it has to satisfy that $a*e=a+e-a\times e=a$. This implies that $e=a\times e$. Suppose $e\ne 0$, then we would get $a=1$, which is impossible since we know that there are lots of real numbers that not equal to zero! So $e=0$.
A: As stated in the other answers, the identity element is $0$. If the goal was to prove or disprove that this is a group, you checked the axioms in an unfortunate order, because inverses don't exist. In particular, $1$ does not have an inverse, because $a\ast 1=1$ for all $a$.
A: Your calculation was good up to
$$
e = ae.
$$
Remember that if $e$ is to be the identity, then you want this equation to hold for all $a$.  If you like, you can rewrite the equation as
$$
0 = (a-1)e,
$$
The only solution is $e = 0$.

By the way, there's a neat way to understand this operation, using the function $r$, defined by $r(x) = 1 - x$.  The equation $c = a * b$ is equivalent to 
$$
\begin{align}
r(c) &= r(a) \cdot r(b) \\
(1-c) &= (1-a) (1-b) \\
1 - c &= 1 - a - b + ab \\
c &= a + b - ab
\end{align}
$$
Luckily $r$ is its own inverse:  $r(r(x)) = x$ for all $x$.  (It's a reflection about the point $x = \tfrac12$.)  So, all of the structure and properties of the usual multiplication on the reals is transferred over (and back) to the $*$ multiplication by $r$.
The multiplicative identity for usual multiplication is $1$, so the multiplicative identity for $*$ multiplication is
$$
r(1) = 1 - 1 = 0.
$$
In algebra, a map such as $r$ that acts as a dictionary, translating from one structure to an equivalent one, is called an isomorphism.
