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Thread: Element of infinite order

  1. #1
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    Element of infinite order

    Let a be a group element that has infinite order. Prove that <a^i> = <a^j> if and only if i = +j or -j.
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  2. #2
    Super Member Gamma's Avatar
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    Suppose \langle a^i \rangle=\{a^{in}|n \in \mathbb{Z} \}= \langle a^j \rangle=\{a^{jn}|n \in \mathbb{Z} \}.

    But if these sets are to be equal, then there must be an integer n such that ni=j and there must be an integer m such that mj=i. That is i|j and j|i. This proves j=+/-i as desired.

    The converse is clear, and i trust you can take care of that.
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