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Math Help - Proving homomorphism

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    Proving homomorphism

    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
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    Quote Originally Posted by Ryan0710 View Post
    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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