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Math Help - Order of a Group with a Unique Subgroup

  1. #1
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    Order of a Group with a Unique Subgroup

    Hello!

    I have a problem I'm working on..
    Given that a finite group G has exactly one nontrivial proper subgroup (call it H),

    then we must prove
    a) G must be cyclic
    b) the order of G is p^2 for a prime p.

    I have shown a) by noting that any element in G either generates
    Case 1:the nontrivial proper subgroup H
    Case 2:the group itself
    So there is at least one element which generates the group (so it is cyclic).

    I am having trouble showing part b), however.

    Any help would be fantastic! and cause me to be very grateful! Thanks!
    Cheers.
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  2. #2
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    Quote Originally Posted by matt.qmar View Post
    Hello!

    I have a problem I'm working on..
    Given that a finite group G has exactly one nontrivial proper subgroup (call it H),

    then we must prove
    a) G must be cyclic
    b) the order of G is p^2 for a prime p.

    I have shown a) by noting that any element in G either generates
    Case 1:the nontrivial proper subgroup H
    Case 2:the group itself
    So there is at least one element which generates the group (so it is cyclic).

    I am having trouble showing part b), however.

    Any help would be fantastic! and cause me to be very grateful! Thanks!
    Cheers.

    So it is cyclic: it there were two primes dividing the order of the group then either Cauchy's Teorem or Sylow's theorems

    would give us at least two different proper non-trivial subgroups, each of order a different prime, so the order of the

    group is divisible only by one prime, say p. It can't be that the order is p, since then there is no proper non-trivial sbgp's at

    all, and it can't be p^n\,,\,n\geq 3 , since then there are at least 2 different proper non-trivial sbgp's, of

    orders p\,,\,p^2 (why??) , so...

    Tonio
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