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Math Help - Equivalence Classes with a Single Element

  1. #1
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    Equivalence Classes with a Single Element

    Hi guys, question here. It's not for a class.

    "Let G be a group. Then the relation a \sim b if and only if a = g^{-1}bg for some g \in G is an equivalence relation on G. Some equivalence classes contain only one element c. Characterize those elements c."

    It's pretty clear that if c is in the center of G, then c must be in a class by itself since b \sim c means b = g^{-1}cg = g^{-1}g c = c so that b = c.

    But could there be more? This question comes after a discussion of the commutator, and the book hasn't even discussed center yet, so I'm trying to figure out what kind of answer the author was after.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by WolfTecc View Post
    It's pretty clear that if c is in the center of G, then c must be in a class by itself

    Coversely, if c\not\in Z_G then, there exists x\neq c such that xc\neq cx which implies c\neq x^{-1}cx. But c\equiv x^{-1}cx so, [c] has at least two elements.
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