According to Mr. Pinter's Abstract Algebra on page-89 he defines isomorphism like this:

*Let $\displaystyle G_1$ and $\displaystyle G_2$ be groups. A bijective function $\displaystyle f:G_1 \to G_2$ with the property that for any two elements $\displaystyle a$ and $\displaystyle b$ in $\displaystyle G_1$,*

$\displaystyle f(ab) = f(a)f(b)$

is called an **isomorphism** from $\displaystyle G_1$ to $\displaystyle G_2$.

If there exists an isomorphism from $\displaystyle G_1$ to $\displaystyle G_2$, we say that $\displaystyle G_1$ is **isomorphic** to $\displaystyle G_2$.

Then in next page Mr. Pinter gives an example of two groups that have different binary operations but are isomorphic.

Does this mean(implicitly) that the binary operations in these two groups have to be different in order for them to be isomorphic? or am I wrong on this?