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Math Help - Conjugacy Classes

  1. #1
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    Conjugacy Classes

    I need to find the conjugacy classes of D5.

    What I have is the following:

    D5 is {e, a, a^2, a^3, a^4, b, ab, a^2b, a^3b, a^4b}
    and a^5 = e, b^2 = e and ba = a^(-1)b

    I am not exactly sure how to find conjugacy classes.

    Thanks for the help.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by page929 View Post
    I need to find the conjugacy classes of D5.

    What I have is the following:

    D5 is {e, a, a^2, a^3, a^4, b, ab, a^2b, a^3b, a^4b}
    and a^5 = e, b^2 = e and ba = a^(-1)b

    I am not exactly sure how to find conjugacy classes.

    Thanks for the help.
    To find conjugacy classes, simply take every element and conjugate it by every other element! This will tell you which elements are conjugate!

    However - there are some `tricks' to save you time. The main one is that conjugacy classes are disjoint, so is g is conjugate to h then g and h are in the same conjugacy class (obviously). So, take an element and work out all its conjugates, by conjugating by every other element. The elements it is conjugate to form its conjugacy class. Do the same to an element not in this conjugacy class, and so you will have two conjugacy classes. Do the same to an element in neither of these two conjugacy classes, and so on...

    Also, don't forget the identity! It always sits in a conjugacy class of its own.
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  3. #3
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    some tips:

    if an element is in the center of G, Z(G) = {x in G: xg = gx, for all g in G}, then gxg^-1 = xgg^=1= x, no matter what g is.

    so every element that commutes with everything has a singleton conjugacy class.

    also, two elements that are conjugate must have the same order: if x^k = e, then (gxg^-1)^k = g(x^k)g^-1 = e.

    another tip: the size of every conjugacy class will divide the order of G.
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Deveno View Post
    also, two elements that are conjugate must have the same order: if x^k = e, then (gxg^-1)^k = g(x^k)g^-1 = e.
    Better than that - if g \sim h then g^n\sim h^n for all n \in \mathbb{Z}! (by ~ I mean conjugate to).
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  5. #5
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    if groups go to jail, do they get conjugate visits?
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