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Conjugacy in algebra

  1. Jun 10, 2008 #1

    tgt

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    conjugacy or a=g^-1bg occur a lot in algebra for a,b,g in G. But why?
     
  2. jcsd
  3. Jun 10, 2008 #2

    matt grime

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    Because it's important?

    Conjugate elements (in a group) have the 'same' properties, essentially. This is because the map

    f_g :G-->G

    f_g(x)=g^{-1}xg

    is an isomorphism. The set of all such f_g, g in G is the group of inner automorphisms. In a lot of cases these are all automorphisms of a finite group; in some cases they are not.

    In linear algebra, conjugate matrices share many properties....

    The number of conjugacy classes of a finite group is the same as the number of simple complex valued representations.

    Shall I go on?
     
  4. Jun 10, 2008 #3

    tgt

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    They first seem a bit weird but now that you mentioned these things, they seem quiet natural.
     
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