I am a math hobbyist reading John Fraleigh's book on Abstract Algebra.

In Section 13 on Homomorphisms Fraleigh gives the following example. { I have put r =2 in his example to make it more tangible and specific)

Let r = 2 Z and let be defined by for all n Z.

For all m,n Z, we have is a homomorphism.

Although Fraleigh expresses this as a homomorphism of it seems to me the target group must be the additive group 2Z = { ..., -4, -2, 0, 2, 4, 6, ...} and not Z as implied by his definition of the homomorphism. [Note that Fraleigh defines a groups homomorphism as being between a group G and a group G' but in the example seems to be referring to a mapping between two sets, with the target group being 2Z which, obviously is a subset of Z]

I would be grateful if someone could clarify this for me

Bernhard