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Is every Subgroup of a Cyclic Group itself Cyclic?

  1. Mar 20, 2013 #1
    1. The problem statement, all variables and given/known data

    Are all subgroups of a cyclic group cyclic themselves?

    2. Relevant equations

    G being cyclic means there exists an element g in G such that <g>=G, meaning we can obtain the whole group G by raising g to powers.

    3. The attempt at a solution

    Let's look at an arbitrary subgroup H= {e, gk_1, gk_2, ... , gk_n}

    We know since subgroups are closed that (gk_i)t is in H for all integers t, and for all i between 1 and n.

    So unless gk_1, gk_2,... all have order 1 (which would mean H ={e}), then by the pidgeonhole principle, we have (gk_i)x = (gk_j)y
    for some i,j and some 0<x<ord(gk_i), 0<y<ord(gk_2).

    WLOG let's say y<x. Then x=yq+r. This implies (gk_i)r=(gk_j).







    ... I'm stuck :frown:
     
  2. jcsd
  3. Mar 20, 2013 #2

    jbunniii

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    Hint: if ##H## is a subgroup of ##G##, then of course all of its elements are of the form ##g^k##. Consider the element of ##H## with the smallest positive exponent. Can you show that it generates ##H##?
     
  4. Mar 20, 2013 #3
    Assume BWOC that g^k doesn't generate H. Then there is an element g^t in H such that k doesn't divide t. But that means t=kq+r where r<k. Which means g^r is in the H but that contradicts k being the smallest power of g in H.



    Thanks!
     
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