Is the set of real numbers a group under the operation of multiplication?

The collection of positive real numbers (and even real numbers without zero) is a group. However, once you append zero, the resulting set is no longer a group for exactly the reason your are suggesting.

One interesting thing about the positive real numbers, $(\mathbb{R}_+,\cdot)$, is that they are isomorphic to the reals with addition, $(\mathbb{R},+)$. This can be seen through the logarithm, $$\log(a\cdot b) = \log(a) + \log(b).$$ Note also that $\log(1)=0$, that is the logarithm identifies the identity elements between the two different groups.

The set $R$ endowed with the operation [ . ] cannot be considered a group, since considering it a group means that every element in $R$ is invertable with respect to [ . ]. Now considering ($R$ , . ) a group, we all know that the inverse of an element say $x$ belonging to $R$ is $\frac{1}{x}$ but since 0 belongs to the group $R$ hence $\frac{1}{0}$ exists in $R$ ( Contradiction ). Meanwhile you can consider (R*,.) a group.