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Math Help - Prove cyclic group in homomorphic image

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    Prove cyclic group in homomorphic image

    Assume that the group G' is a homomorphic image of the group G.

    a. Prove that G' is cyclic if G is cyclic.

    b. Prove that o(G') divides o(G), whether G is cyclic or not
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    MHF Contributor Drexel28's Avatar
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    So, for the sake of notational convenience let's call our homomorphism \theta:G\to G'.

    Also, I'll assume these are finite groups.

    Quote Originally Posted by rainyice View Post
    Assume that the group G' is a homomorphic image of the group G.

    a. Prove that G' is cyclic if G is cyclic.
    So, let G=\langle g\rangle. We claim that G'=\langle \theta(g)\rangle. To see this let g'\in G be arbitrary. By assumption g'=\theta(h) for some h\in G. But, that means that h=g^n for some n and so g'=\theta(h)=\theta(g^n)=\theta(g)^n.

    Work with that.

    b. Prove that o(G') divides o(G), whether G is cyclic or not
    How much group theory do you know? You should know by the FIT that G/\ker\theta\cong G' and so |G'|=\left|G/\ker\theta\right|=\frac{|G|}{|\ker\theta|}\implies  \frac{|G|}{|G'|}=|\ker\theta|
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    Quote Originally Posted by Drexel28 View Post
    So, for the sake of notational convenience let's call our homomorphism \theta:G\to G'.

    Also, I'll assume these are finite groups.



    So, let G=\langle g\rangle. We claim that G'=\langle \theta(g)\rangle. To see this let g'\in G be arbitrary. By assumption g'=\theta(h) for some h\in G. But, that means that h=g^n for some n and so g'=\theta(h)=\theta(g^n)=\theta(g)^n.

    Work with that.



    How much group theory do you know? You should know by the FIT that G/\ker\theta\cong G' and so |G'|=\left|G/\ker\theta\right|=\frac{|G|}{|\ker\theta|}\implies  \frac{|G|}{|G'|}=|\ker\theta|
    Thank you for explaining in detail. I learned the definition of group, subgroup, and cyclic group. Now, I am in the session of Isomorphism and Homomorphism. For the Kernal you just showed me, I have not learned yet.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by rainyice View Post
    Thank you for explaining in detail. I learned the definition of group, subgroup, and cyclic group. Now, I am in the session of Isomorphism and Homomorphism. For the Kernal you just showed me, I have not learned yet.
    "Kernel"

    What do you know? What are you currently studying? Do you have any idea of how to show this without using the FIT?
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    Quote Originally Posted by Drexel28 View Post
    "Kernel"

    What do you know? What are you currently studying? Do you have any idea of how to show this without using the FIT?
    I just learned Kernel today. Kernel # = {x E G | #(x) = e'} but i haven't hear about FIT
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by rainyice View Post
    I just learned Kernel today. Kernel # = {x E G | #(x) = e'} but i haven't hear about FIT
    Take a look.
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