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## Homework Statement

Consider the mapping [itex] \phi \colon (\mathbb{Z} , + ) \to (\mathbb{Z}, *) [/itex] such that [itex] \phi (a) = a+2[/itex]. Define * such that [itex] \phi [/itex] is a homomorphism. For [itex] (\mathbb{Z}, *) [/itex], define the identity element.## Homework Equations

## The Attempt at a Solution

Well, first I thought I was oversimplifying this problem. I know that since we have a homomorphism, the identity of [itex] (\mathbb{Z} , \plus ) [/itex] will map to the identity element of our image. The identity of [itex] (\mathbb{Z} , \plus ) [/itex] will be 0, so [itex] \phi (0) = 1 = e^{\prime} [/itex]. But that seemed too easy.So my question, do I have to break out properties of the homomorphism, namely that [itex] \phi (a+ b) = \phi (a) * \phi (b) = (a+2) * (b+2) [/itex] somehow.

I appreciate any help, thanks!