1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    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. 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


    User Avatar
    Homework Helper
    Gold Member

    Yes, it all looks OK to me.
  6. Mar 15, 2012 #5
    Ok, thanks!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Prove map σ:y→xyx⁻¹ is bijective
  1. Proving Bijections (Replies: 1)

  2. Prove a bijection. (Replies: 5)

  3. Proving bijection (Replies: 4)

  4. Bijective Maps (Replies: 9)