Need help proving homomorphism.
Show that the mapping W:Z6 ->Z24 given by W([B]x/B])=4[B]x/B] is well defined and is in fact a homomorphism. Find the image and kernal of W
This is well-defined. And $\displaystyle W([x_1])+W([x_2]) = 4[x_1]+4[x_2] = 4[x_1+x_2] = W([x_1]+[x_2])$.
The kernel are all $\displaystyle [x]$ so that $\displaystyle 4[x] = [0]\implies 4x\equiv 0(\bmod 24)$. Which happens when $\displaystyle x$ is multiple of $\displaystyle 6$, so, $\displaystyle [6]=[0]$.
The image are all $\displaystyle [y]\in \mathbb{Z}_{24}$ so that $\displaystyle [4x] = [y]$ which happens when $\displaystyle 4x\equiv y(\bmod 24)$.
Note for this equation to be solvable we require $\displaystyle 4=\gcd(4,24)$ to divide $\displaystyle y$. Thus, $\displaystyle y=[4]$.