It would be natural to require also that φ φ respect identities, ⁠ φ φ ( 1 G ) = 1 H ⁠, and inverses, φ φ ( a − − 1 ) = φ φ ( a ) − − 1 for all a in ⁠ G ⁠. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.