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Math Help - Group having 3 proper subgroups

  1. #1
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    Group having 3 proper subgroups

    What can we say about a group which has exactly 3 proper subgroups?
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  2. #2
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    Quote Originally Posted by Chandru1 View Post
    What can we say about a group which has exactly 3 proper subgroups?


    Since any non-trivial group always has two trivial subgroups (the trivial one and the whole group), we're looking for a group with one single non-trivial subgroup...hint: how many primes can divide the group's order?

    Tonio
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by tonio View Post
    Since any non-trivial group always has two trivial subgroups (the trivial one and the whole group), we're looking for a group with one single non-trivial subgroup...hint: how many primes can divide the group's order?

    Tonio
    No - proper means it is properly contained in it. That is to say, H \lneq G. So we are looking for a group with precisely two non-trivial subgroups (although I think your hint is still the way to go).
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    Quote Originally Posted by Chandru1 View Post
    What can we say about a group which has exactly 3 proper subgroups?
    As tonio said, a cyclic group of order p^3 can be one of examples, where p is a prime number.

    See my previous post here .
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by TheArtofSymmetry View Post
    As tonio said, a cyclic group of order p^3 can be one of examples, where p is a prime number.

    See my previous post here .
    As a warning to the OP, this does not classify them all. Another example would be V_4, the klein 4-group (or, more generally, the cross-product of two groups of prime order).
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  6. #6
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    Quote Originally Posted by Swlabr View Post
    No - proper means it is properly contained in it. That is to say, H \lneq G. So we are looking for a group with precisely two non-trivial subgroups (although I think your hint is still the way to go).

    Yes. I oversaw the word "proper" in the OP, but still the hint remains...though nevertheless there are OTHER examples: for example, a cyclic group of order 10...

    Tonio
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