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Exercise from Serre

  1. Aug 2, 2007 #1
    This is an exercise from a book from Serre called Trees. Given the group

    G = < a, b, c | bab−1 = a2, cbc−1 = b2, aca−1 = c2 >

    I have to prove G = 1.

    I don't have a clue. Of course G' = G (commutator subgroup equals the group itself) but I don't know what to deduce from that. Another first step could be to prove that the orders of a, b and c are finite. But I do not even know how to that.

    If anyone could put me in the right direction i would be very grateful.

    edit: Im gonna try to use Todd Coxeter coset enumeration.
     
    Last edited: Aug 2, 2007
  2. jcsd
  3. Aug 2, 2007 #2
    I doubt wetter ToddCoxeter CE will help me :frown:
     
  4. Aug 2, 2007 #3

    matt grime

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    Since the question uses a,b,c and it's Serre, let's assume that the fact there are 3 generators is important.

    The only thing I can think of doing is considering products like abc or bac etc and simplifying in two ways until ended up with something that points towards the statement a=a^2, or similar.

    (No, I\ve not solved this - I'll get a pen and paper and think about it some more later)
     
  5. Aug 3, 2007 #4
    well, if you can prove all the 3 generators are of finite order than I can finish this question. Because Suzuki has this exercise in his book where he assumes G is finite and he also gives lots of hints.

    Here however I can not a priori assume G is finite.
     
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