Answer to Question #4997 in Abstract Algebra for Annemarie

Question

Consider the function f : Z4 → Z12 defined by f(x) = 3x. Show that f(x) does
NOT preserve multiplication.

Answer 1

f:Z₄ --> Z₁₂, f(x)=3x
We will show that this function does not preserve multiplication. we can consider x,y from Z₄ such that

let x=1, y=3 so we have:
f(1*3)=f(3)=9 (modulo 12)
f(1)*f(3)=3*9=27=3( modulo 12)
But 9 is not equal to 3 in Z₁₂. So f does not preserve multiplication.