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!

Simple groups

  1. Feb 3, 2008 #1
    [SOLVED] simple groups

    1. The problem statement, all variables and given/known data
    T or F: All nontrivial finite simple groups have prime order.


    2. Relevant equations



    3. The attempt at a solution
    I want to say yes with Lagrange's Theorem, but I am not sure that applies...
     
  2. jcsd
  3. Feb 3, 2008 #2

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    It's false. A counterexample is going to be necessarily non-abelian (why?).
     
  4. Feb 3, 2008 #3
    I think I see. The reason is that the converse of Lagrange's theorem is false, right? I mean that if a group has order a and b divides a, there is not necessarily a subgroup of G that has an order of a.

    The classic example is A_4. I believe this has no proper subgroups although it has order 12.

    I am not exactly sure about the answer to your "why". Is it because any finite abelian group can be written as Z_p^r cross ... and you just multiply the p^r s to get the order of the group?
     
  5. Feb 3, 2008 #4

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    Yup - that the converse to Lagrange is false in general is basically why this statement is false. (Incidentally, the converse is true for abelian groups. Thus, the only finite simple abelian groups are those of prime order. And, in fact, any simple abelian group is finite.)
     
  6. Feb 3, 2008 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    ehrenfest, you do know that 'simple group' means 'no normal subgroups', right? A_4 is a bad classic example. It has a both proper subgroups and a normal subgroup. I think what morphism is getting at is that ANY subgroup of an abelian group is normal. And Lagrange's theorem gives you one automatically if the order of the group is not prime.
     
    Last edited: Feb 3, 2008
  7. Feb 3, 2008 #6
    umm. so what is a counterexample then?
     
  8. Feb 3, 2008 #7

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    A very classical and popular one: A_5. This is the smallest finite simple non-abelian group.
     
  9. Feb 3, 2008 #8

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    I should expand a bit on my comments in #4. Suppose the converse to Lagrange's theorem held for all types of finite groups. Say we're given a nontrivial finite group G whose order is not prime. Let p be the smallest prime divisor of |G|. Then G must have (since we're assuming Lagrange's converse holds) a subgroup H of order |G|/p. This subgroup has index p in G, and is thus normal. (Here I'm using the following nifty fact: If H is a subgroup of a finite group G, and [G:H] is the smallest prime divisor of G, then H is normal.) Moreover, H is nontrivial and not the entire group. So G cannot be simple.
     
    Last edited: Feb 3, 2008
  10. Feb 3, 2008 #9

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I suppose. But "supposing the converse to Lagrange's theorem held" is sort of ultra-hypothetical.
     
  11. Feb 3, 2008 #10

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    Definitely - and that was the point of my post - just hypothesizing about why a counterexample should exist. :smile: I guess I should have also explicitly mentioned that A_4 is not a counterexample to the statement in the OP, in post #4.
     
  12. Feb 3, 2008 #11

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Too bad counterexamples do exist. Otherwise we could have gotten over this group classification problem before they invented computers. :)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Simple groups
  1. Simple Group Problem (Replies: 2)

Loading...