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Math Help - Cyclic group

  1. #1
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    Cyclic group

    Prove that all subgroup of a cyclic group is cyclic.

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  2. #2
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    Let (G,.) be a cyclic group, H a subgroup of G, and \alpha \in G such that <\alpha >=G.
    Let \phi : \mathbb{Z} \rightarrow G : n \mapsto \alpha^{n} be the only morphism of groups such that \phi(1)=\alpha.

    Then \phi^{-1}(H) is a subgroup of (\mathbb{Z},+), so it's a cyclic group.
    So \phi(\phi^{-1}(H))=H (because of surjectivity) is a cyclic group.


    Why, if K is a cyclic group, L another group, and \psi a morphism of groups between K and L, then \psi(K) is cyclic?

    Let a be a generator of K, and b an element of \psi(K). There is a n \in \mathbb{N} such that b=\psi(a^{n})=\psi(a)^{n}. Thus <\psi(a)>=\psi(K)
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  3. #3
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    Quote Originally Posted by roporte View Post
    Prove that all subgroup of a cyclic group is cyclic.

    Thanks!
    Let H be a subgroup. And a generate the large group G. Choose n\in \mathbb{Z}^{\times} to be the smallest such that a^n\in H. Now prove that \left< a^n \right> = H.
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