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Every infinite cyclic group has non-trivial proper subgroups

  1. May 4, 2017 #1
    1. The problem statement, all variables and given/known data
    Every infinite cyclic group has non-trivial proper subgroups

    2. Relevant equations


    3. The attempt at a solution
    I know that if we have a finite cyclic group, it only has non-trivial proper subgroups if the order of the group is not prime. But I'm not sure how to make this argument with infinite groups
     
  2. jcsd
  3. May 4, 2017 #2

    fresh_42

    Staff: Mentor

    What is a infinite cyclic group and how does it differ from a finite one?
     
  4. May 4, 2017 #3
    Doesn't it differ by the fact that there exists no integer ##n## for which ##a^n=e##, where ##a## is an element of the group besides e, the neutral element?
     
  5. May 5, 2017 #4

    fresh_42

    Staff: Mentor

    Yes. So we have ##C = \langle a^n\,\vert \,n \in \mathbb{Z} \rangle = \{\ldots , a^{-2},a^{-1},e,a,a^2,\ldots\}##.
    Non-trivial subgroup ##U## means ##\{e\}\subsetneq U \subsetneq C##.
    So what can we conclude from ##\{e\}\subsetneq U##?
    What could be a subgroup?
     
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