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Math Help - Group of prime power order with unique subgroup

  1. #1
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    Group of prime power order with unique subgroup

    If a group of order p^m, where p is an odd prime, contains a unique subgroup of order p^s, for some 0 <s <m ,
    then G must be cyclic.

    How to prove this ?. Plz help.
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  2. #2
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    Re: Group of prime power order with unique subgroup

    Suppose it were NOT cyclic. Then there exist elements a and b such that a is not a power of b and b is not a power of a. Show that the set of powers of a form a subgroup and the set of powers of b form a different subgroup.
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  3. #3
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    Re: Group of prime power order with unique subgroup

    but here the condition is "for some 0 < s < m " not "for all 0 < s < m ".
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  4. #4
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    Re: Group of prime power order with unique subgroup

    Hi,
    I've attached a solution. Initially, I was going to provide all the details, but then I realized I was in essence proving the theorem quoted first. So I just quoted the theorem with a reference.

    Group of prime power order with  unique subgroup-mhfgroups12a.png

    Group of prime power order with  unique subgroup-mhfgroups12b.png
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