Proving homomorphism

**Ryan0710** · May 8th 2008, 06:33 AM

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

**ThePerfectHacker** · May 8th 2008, 07:22 AM

Originally Posted by Ryan0710

Need help proving homomorphism.
Show that the mapping W:Z6 ->Z24 given by W(x)=4x is well defined and is in fact a homomorphism. Find the image and kernal of W

This is well-defined. And $W([x_1])+W([x_2]) = 4[x_1]+4[x_2] = 4[x_1+x_2] = W([x_1]+[x_2])$ .
The kernel are all $[x]$ so that $4[x] = [0]\implies 4x\equiv 0(\bmod 24)$ . Which happens when $x$ is multiple of $6$ , so, $[6]=[0]$ .
The image are all $[y]\in \mathbb{Z}_{24}$ so that $[4x] = [y]$ which happens when $4x\equiv y(\bmod 24)$ .
Note for this equation to be solvable we require $4=\gcd(4,24)$ to divide $y$ . Thus, $y=[4]$ .

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