Prove map σ:y→xyx⁻¹ is bijective

  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. LCKurtz

    LCKurtz 8,186
    Homework Helper
    Gold Member

    Instead of showing ##y_1\ne y_2 \to xy_1x^{-1}\ne xy_2x^{-1}## try showing the contrapositive.
     
  4. 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. LCKurtz

    LCKurtz 8,186
    Homework Helper
    Gold Member

    Yes, it all looks OK to me.
     
  6. Ok, thanks!
     
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