Thread: Left and right cosets (modern Algebra)

1. Left and right cosets (modern Algebra)

Let H be a subgroup of G, and let S={left cosets of H in G}, and T={Right cosets of H in G}. Now Define f: S-->T, by the following function f(aH)=H(a^-1)
Prove that f is well define!!!

All I know is "A well defined set is that if aH = bH then f(aH) = f(bH)"

2. Re: Left and right cosets (modern Algebra)

To show a relation f is "well defined", you need to verify if x=z, then f(x)=f(z); i.e. f is single valued. So let aH and bH be left cosets of H in G. Then aH=bH iff b-1a is in H iff Ha-1=Hb-1. This actually shows more: f is a 1:1 function. Furthermore f is onto (you can do this), and so the number of left cosets is the number of right cosets.

3. Re: Left and right cosets (modern Algebra)

I started with aH=bH and did left cancellation. I was able to get to (b^-1)aH=H. How can I proceed?

4. Re: Left and right cosets (modern Algebra)

Do the same thing on the right:
$Ha^{-1}=Hb^{-1}$ iff $Ha^{-1}b=H$