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Math Help - Prove that any cyclic group of finite order n is isomorphic to Zn under addition.

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    Prove that any cyclic group of finite order n is isomorphic to Zn under addition.

    Prove that any cyclic group of finite order n is isomorphic to Zn under addition.

    Should I draw two tables to indicate that? Like ... the first one is cyclic group table G and the second one is the Zn table H, then I linked it G -> f -> H where f is the transformation.
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    Quote Originally Posted by rainyice View Post
    Prove that any cyclic group of finite order n is isomorphic to Zn under addition.

    Should I draw two tables to indicate that? Like ... the first one is cyclic group table G and the second one is the Zn table H, then I linked it G -> f -> H where f is the transformation.


    Too many questions , too little work shown: what've you done to solve these questions? Where are you stuck?

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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by rainyice View Post
    Prove that any cyclic group of finite order n is isomorphic to Zn under addition.

    Should I draw two tables to indicate that? Like ... the first one is cyclic group table G and the second one is the Zn table H, then I linked it G -> f -> H where f is the transformation.
    Hint : map a generator to a generator, and extend the map to a homomorphism in the only possible way; then show that it's an isomorphism.
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    Question Working on the same problem

    Quote Originally Posted by rainyice View Post
    Prove that any cyclic group of finite order n is isomorphic to Zn under addition.

    Should I draw two tables to indicate that? Like ... the first one is cyclic group table G and the second one is the Zn table H, then I linked it G -> f -> H where f is the transformation.
    I don't think drawing tables qualifies as a proof (I'm working on the same problem). Also, you weren't told what operation "G" was under, so you wouldn't be able to make a table for cyclic group G.

    I believe we have to show that some generator <x> of G maps into some generator <a> of Z. To show isomorphism, we have to satisfy two conditions. To show
    φ (some function) is a one-to-one correspondence from G to Z, we need to show that every element in G maps into every element in Z. For this one, I'm assuming since both groups are of the same order (we stated that cyclic group G is of order n and it is given that Z is of order n), there is automatically a one-to-one correspondence between the two groups (I'm not sure though).

    I'm still working on showing the second condition of isomorphism.

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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by MissMousey View Post
    I don't think drawing tables qualifies as a proof (I'm working on the same problem). Also, you weren't told what operation "G" was under, so you wouldn't be able to make a table for cyclic group G.

    I believe we have to show that some generator <x> of G maps into some generator <a> of Z. To show isomorphism, we have to satisfy two conditions. To show
    φ (some function) is a one-to-one correspondence from G to Z, we need to show that every element in G maps into every element in Z. For this one, I'm assuming since both groups are of the same order (we stated that cyclic group G is of order n and it is given that Z is of order n), there is automatically a one-to-one correspondence between the two groups (I'm not sure though).

    I'm still working on showing the second condition of isomorphism.

    Some people call me the gangster of love.
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