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Consider θ:Z -> Z is a mapping where θ(n) = n^3 and it's homomorphism under multiplication. In this case, it's not a homomorphism under addition.
So my question is this. In general, if we show that a group is homomorphic under multiplication, does this imply that it is not under addition and vice versa?
So my question is this. In general, if we show that a group is homomorphic under multiplication, does this imply that it is not under addition and vice versa?