Quantcast

A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

Mr.Math

  • 2 years ago

A problem in group theory. Prove that a group has exactly \(3\) subgroups if and only if it's a finite cyclic group of order \(p^2\).

  • This Question is Closed
  1. nbouscal
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Well, we know that the order of a subgroup must divide the order of the group. So, if we have a finite cyclic group of order \(p^2\), its subgroups will necessarily have order \(1, p, \text{or } p^2\). We can prove that there is only one possible group with each of these orders somehow, something to do with the group of order 1 being the trivial subgroup and finite cyclic groups of order \(p\) having only one isomorphism class for a given \(p\), I think.

  2. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I know how to prove that a finite cyclic group of order \(p^2\) has exactly \(3\) subgroups. This follows directly from the Fundamental Theorem of Finite Abelian Groups. But I'm not sure how to show that it's the only group (up to isomorphism).

  3. nbouscal
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I haven't actually even learned the FTowlet yet, I've been putting group theory on hold while doing some analysis work. To go the other direction with this proof though I think one would want to look at prime factorizations with 3 elements to show that it only works for \(p^2\)? Dunno.

  4. nbouscal
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Haha, censored abbreviation, whoops.

  5. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    That sounds like a good idea.

  6. nbouscal
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Intuitively, it is obvious that a number must be a squared prime to have a 3-element prime factorization. Rigorously, I'm not entirely sure how to approach showing that.

  7. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Yep!

  8. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I think I know how.

  9. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Let's denote this group by \(G\) where \(|G|=n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}\). Then we know that \(G\) and \(\{e\}\) are subgroups. We want \(n\) to be such that it has only one factor other than itself and \(1\). This can only be if there exists a unique \(p_i|n\) and \(p_i\ne n\). So \(n=p_i^{\alpha_i}\) for some \(1\le i \le r\). Now if \(\alpha_i\ge 3\), then \(p^3|n\) and \(p^2|n\), which contradicts our hypothesis. So \(\alpha_i=2\).

  10. nbouscal
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Sounds right to me. Thanks for reminding me that I need to get back to studying group theory, I've been putting it off for the last couple weeks :P

  11. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    You're welcome! And thanks for your help!

  12. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.