homomorphism

If \(A\) and \(B\) are two groups, then \(\phi\) is a group homomorphism if \(\phi(a)\cdot \phi(b) = \phi(a\cdot b)\) where \(\cdot\) is the group operation

That is, the mapping preserves the group operation. I can perform the group operation and then the mapping. Or, the mapping and then the group operation in the new space.