Show that phi Z12 rightarrow Z10 both groups are under addi

Show that phi : Z_12 rightarrow Z_10 (both groups are under addition) defined via phi(x) = 3x is not a group homomorphism.

Solution

A homomorphism is a map that preserves selected structure between two algebraic structures, with the structure to be preserved being given by the naming of the homomorphism.

we are taking phi(x) = g(x)

A function which preserves addition should have this property: f(a + b) = f(a) + f(b). For example, g(x) = 3x is one such homomorphism, since f(a + b) = 3(a + b) = 3a + 3b = f(a) + f(b). Note that this homomorphism maps the Z12 to Z10 back into themselves.