For two groups to be isomorphic is it a requirement that binary operations in the two groups have to be different?

Is there a situation where binary operations between two isomorphic groups are same?

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

Let and be groups. A bijective function with the property that for any two elements and in ,

is called anisomorphismfrom to .

If there exists an isomorphism from to , we say that isisomorphicto .

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?