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Prove map σ:y→xyx⁻¹ is bijective

  1. Mar 15, 2012 #1
    1. Let G be any group and x∈G. Let σ be the map σ:y→xyx⁻¹. Prove that this map is bijective.
    It seems to be written strangely, since it never really says anywhere that y is in G, but I guess that must be an assumption.


    2. bijective=injective+surjective.
    in order to prove injective, we need to show that y1≠y2→xy1x⁻¹≠xy2x⁻¹
    and in order to prove surjective, we need to show that for every g in G, there exists a y in G such that xyx^-1=g.




    3. I think that I can say: Let y=x^-1gx. Then xyx-1=g and we are done for surjective.
    I don't really know how to "show" injective, since it seems obvious.
     
  2. jcsd
  3. Mar 15, 2012 #2

    LCKurtz

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    Instead of showing ##y_1\ne y_2 \to xy_1x^{-1}\ne xy_2x^{-1}## try showing the contrapositive.
     
  4. Mar 15, 2012 #3
    So I show that xy1x-1=xy2x-1→y1=y2 by simply left-multiplyng both sides by x-1 and right-multiplying both sides by x? Is that too simple?

    Also, does my thinking on surjective work?
     
  5. Mar 15, 2012 #4

    LCKurtz

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    Yes, it all looks OK to me.
     
  6. Mar 15, 2012 #5
    Ok, thanks!
     
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