TITLE: Does the operation of a group matter when talking about isomorphism?
QUESTION [7 upvotes]: When we say two groups are isomorphic to each other, do we have to specify the operations on each group? Does this matter? For example, here is a problem from the book 'Algebra and Geometry' by A.F.Beardon:

Let $G$ be the group of real $2\times 2$ matrices of the form
  $$M(a)= \begin{pmatrix}
         a & a \\
         a & a \\
        \end{pmatrix}
$$
  where $a \neq 0$.
Show that $G$, under the usual multiplication of matrices, is isomorphic to $\mathbb R^*$, the group of non-zero real numbers.

I solved this problem by defining an isomorphism $$\theta(M(a))=2a$$
which does not involve any specification of the operation on group $\mathbb R^*$.
So my question is : When we talk about the isomorphism between two groups, is it important to specify also the operations defining the two groups?

REPLY [8 votes]: You assert that $\theta$ is an isomorphism. But, how did you check that this assertion is true? What you had to do, in order to verify the definition of isomorphism, is to verify the equation $\theta(M(a) \cdot M(b)) = 2a \cdot 2b$. That "$\cdot$" character on the left hand side is the group operation in $G$, and the "$\cdot$" character on the right hand side is the group operation in $\mathbb{R}^*$. So it does involve specifying the group structure on the source and the target.