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Math Help - Conjugacy Classes

  1. #1
    Super Member Deadstar's Avatar
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    Conjugacy Classes

    Find the conjugacy classes in D_4.

    Is there an easier way to do this than cutting out a square and checking every possibility?
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  2. #2
    Super Member Gamma's Avatar
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    slightly quicker

    You know in general that every element of the center is in its own conjugacy class. When calculating orbits of group elements it is also useful to note you will not get anything new from anything in the centralizer of the element. Lastly, keep in mind that conjugacy classes is an equivalence relation, so hopefully those things can trim your work down a bit.
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  3. #3
    Senior Member JaneBennet's Avatar
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    Let \rho be an element of order 4 in D_4 (i.e. a \tfrac\pi2 rotation).

    Then Z(D_4)=\{1,\rho^2\} so two of the conjugacy classes are \{1\},\ \{\rho^2\}.

    Now \rho and \rho^3 are the only elements of D_4 of order 4. Since elements in the same conjugacy class have the same order, and since \rho and \rho^3 donít belong to the centre (and so donít form their own singleton conjugacy classes), \{\rho,\rho^3\} must be another conjugacy class.

    This leaves you with just 4 elements (the reflections) to check.
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