$$\phi(ab) = \phi(g^k \cdot g^l) = \phi(g^k+l) = k+l + n\mathbbZ = (k + n\mathbbZ) + (l + n\mathbbZ) = \phi(a) + \phi(b).$$
The Crucible of Group Theory: A Comprehensive Guide to Dumm it and Foote, Chapter 4 abstract algebra dummit and foote solutions chapter 4
Avoid sites like Chegg or Course Hero for D&F. Many posted solutions contain critical errors, especially in group actions and Sylow proofs. $$\phi(ab) = \phi(g^k \cdot g^l) = \phi(g^k+l) =
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