1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proof that S is a generating set.

  1. Feb 24, 2013 #1
    1. The problem statement, all variables and given/known data
    I have recently started a new course in Algebra. I have to proof that if S is a subset of a finite group G, with an order greater than half the order of G, S is a generating subset for G.


    2. Relevant equations
    I haven't had cosets nor Lagrange theorem so I suppose I should try to prove it from scratch.


    3. The attempt at a solution
    I have put my mind to the problem for a while now but didnt come up with any meaningful clues. I would be very thankful for any tips.
     
  2. jcsd
  3. Feb 24, 2013 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Let H be the group generated by S. That's a subgroup of G and it has at least as many elements as S. Suppose H isn't the whole group G. Then there is an element x of G that's not in H. Can you show H and xH are disjoint sets and they have the same number of elements? Sure, xH is a coset. But you don't have to know that to do the proof.
     
    Last edited: Feb 24, 2013
  4. Feb 24, 2013 #3
    Thanks for the fast reply ! : ) Thank you I will work with that.
     
  5. Feb 24, 2013 #4
    Ok so I know that H and xH must be disjoined sets (easy to proof), they have the same number of elements and both must be in G. But since the order of H is greater than half the order of G, H and xH unified would have more elements than G which is a contradiction. Therefore H must be the whole group G. Is this right?
     
  6. Feb 24, 2013 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Absolutely correct.
     
  7. Feb 24, 2013 #6
    Ok thanks again for your help : ).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Proof that S is a generating set.
  1. Generating Sets (Replies: 1)

  2. Set Proof (Replies: 14)

Loading...