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Math Help - sub group of cyclic group

  1. #1
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    sub group of cyclic group

    Prove any subgroup (H) of a cyclic group (G) is itself cyclic

    Approach:
    Let G = (a)
    Assume H is non-trivial
    There is a a^k such that k is the smallest +ve power among all a^i that are in H. Well order principle of natural numbers ensures there will always be a unique such k as long as H is non-trivial
    Claim H = (a^k)
    Let a^i \in H
    i = mk+r where r<k
    obviously a^r  \in H (closure property of H)
    Thus r can't be a +ve number (as k is the smallest). Thus r=0.
    Hence a^i = (a^k)^m
    And, H = (a^k) (By definition)

    Is this proof 'rigorous' enough?
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  2. #2
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    Quote Originally Posted by aman_cc View Post
    Prove any subgroup (H) of a cyclic group (G) is itself cyclic

    Approach:
    Let G = (a)
    Assume H is non-trivial
    There is a a^k such that k is the smallest +ve power among all a^i that are in H. Well order principle of natural numbers ensures there will always be a unique such k as long as H is non-trivial
    Claim H = (a^k)
    Let a^i \in H
    i = mk+r where r<k
    obviously a^r  \in H (closure property of H)
    Thus r can't be a +ve number (as k is the smallest). Thus r=0.
    Hence a^i = (a^k)^m
    And, H = (a^k) (By definition)

    Is this proof 'rigorous' enough?

    Rigorous and nice enough. Mention though something about dividing an integer by another integer, euclidean property or something like that.

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    Rigorous and nice enough. Mention though something about dividing an integer by another integer, euclidean property or something like that.

    Tonio

    Thanks Tonio. Will do that.
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