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Math Help - Order of elements in a group

  1. #1
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    Order of elements in a group

    This probably just a simple manipulation problem but I keep going in circles. It's from Introduction to Algebra by Cameron
    A group which contains elements a,b,c,d,e (none equal to the identity) such that ab=c, bc=d, cd=e, de=a, ea=b. Find the orders of a,b,c,d,e.
    I found the 'squares' and 'cubes' of the elements, but it starts to get ugly with higher powers. I know I'm probably suppose to be looking for when a pattern repeats for the powers, but I always end up with a horrible expression or back where I started. Any help pointing me in the right direction, please?
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  2. #2
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    Quote Originally Posted by bleys View Post
    This probably just a simple manipulation problem but I keep going in circles. It's from Introduction to Algebra by Cameron
    A group which contains elements a,b,c,d,e (none equal to the identity) such that ab=c, bc=d, cd=e, de=a, ea=b. Find the orders of a,b,c,d,e.
    I found the 'squares' and 'cubes' of the elements, but it starts to get ugly with higher powers. I know I'm probably suppose to be looking for when a pattern repeats for the powers, but I always end up with a horrible expression or back where I started. Any help pointing me in the right direction, please?
    Interesting problem! I started by expressing everything in terms of a and b:

    c=ab,
    d = bc=bab,
    e=cd = ab^2ab,
    a = de = babab^2ab,
    b = ea = ab^2aba.

    From the last two of those relations it follows that a^2 = babab^2aba = bab^2.\qquad({}^*)

    After a good deal of experimentation I came across this:

    edcbab = edcbc = edcd = ede = ea = b. Multiplying on the right by b^{-1}, you see that edcba = 1, the identity element of the group. (I have to use 1 for the identity element, because e already represents a group element.)

    Therefore, writing the elements in terms of a and b, 1 = edcba = ab^2ab^2abab^2a = ab(bab^2)a(bab^2)a = aba^6 (from (*)). Thus b = a^{-7} and in particular b commutes with a. You can then easily check that a^{11} = 1, and the other elements b, c, d, e also have order 11.
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  3. #3
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    Thank you very much for replying! I also found out a^2 =  bab^2 but would have never even guessed that second expression for b. Thank you for your help!
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