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Thread: conjuacy class

  1. #1
    Junior Member
    Mar 2008

    conjuacy class

    Need some help on the following:

    If we let [x] be conjugacy class of x in group H. Show that$\displaystyle \mid\left [ h\right ] \mid = \mid\left[ h^{-1} \right ] \mid$ for any element h in H

    Thank you for help
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  2. #2
    Global Moderator

    Nov 2005
    New York City
    Let $\displaystyle G$ be group. To show that $\displaystyle | [h] | = | [h^{-1}]|$ you need to show there is a one-to-one correspondence $\displaystyle \phi: [h]\mapsto [h^{-1}]$. Let $\displaystyle x\in [h]$ then it means $\displaystyle x=ghg^{-1}$ for some $\displaystyle g\in G$. Define $\displaystyle \phi(x) = g x^{-1} g^{-1}$. Show this mapping is well-defined. Now show it is a bijection.
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