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Prove map σ:y→xyx⁻¹ is bijective 
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#1
Mar1512, 08:25 PM

P: 49

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 xyx1=g and we are done for surjective. I don't really know how to "show" injective, since it seems obvious. 


#2
Mar1512, 09:19 PM

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



#3
Mar1512, 09:28 PM

P: 49

So I show that xy1x1=xy2x1→y1=y2 by simply leftmultiplyng both sides by x1 and rightmultiplying both sides by x? Is that too simple?
Also, does my thinking on surjective work? 


#4
Mar1512, 09:32 PM

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Prove map σ:y→xyx⁻¹ is bijective
Yes, it all looks OK to me.



#5
Mar1512, 09:33 PM

P: 49

Ok, thanks!



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