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Math Help - Prove that no proper nontrivial subgroup of Z is finite.

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    Prove that no proper nontrivial subgroup of Z is finite.

    Prove that no proper nontrivial subgroup of Z is finite. (Z is the set of integers)
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    Member ModusPonens's Avatar
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    Re: Prove that no proper nontrivial subgroup of Z is finite.

    Hint: are the subgroups of Z cyclic?
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    MHF Contributor Drexel28's Avatar
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    Re: Prove that no proper nontrivial subgroup of Z is finite.

    Quote Originally Posted by dori1123 View Post
    Prove that no proper nontrivial subgroup of Z is finite. (Z is the set of integers)
    Another way to think about it is that if G is some finite subgroup of \mathbb{Z} and g\in G then by Lagrange's theorem you know that |G|g=0. Now tell me, for what g\in\mathbb{Z} does that equation have a solution?
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Prove that no proper nontrivial subgroup of Z is finite.

    Another way: if H\neq \{0\} is a subgroup of \mathbb{Z} , use the Euclidean Division to prove that H=(m) for some m\in \mathbb{Z}-\{0\}.



    P.S. Of course, this way provides unnecessary information with respect to the initial question.
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