if H is a subgroup of G let a~b iff a^(-1)*b is in H
if H is a subgroup of G let a~b iff a*b^(-1) is in H

are these the same two equivalence relations? if so prove, if not give counter example.

Question

if H is a subgroup of G let a~b iff a^(-1)*b is in H
if H is a subgroup of G let a~b iff a*b^(-1) is in H

are these the same two equivalence relations? if so prove, if not give counter example.

zzr0ck3r · Answer

I don't think they are but I cant think of a good counter example.

kinggeorge · Answer

I couldn't resist giving this a try. Are $a,b\in H$ or are they in $G$?

zzr0ck3r · Answer

G

kinggeorge · Answer

Then I'm pretty sure these are not the same equivalence relations. To find the counterexample, I think using $S_4$ would work, although we might need to go to $S_5$.

kinggeorge · Answer

If $a=(123)$ and $b=(234)$, then $a^{-1}b=(134)$. But $ab^{-1}=(124)$. So if we let $H=\langle(134)\rangle$, then $a^{-1}b\in H$, but $ab^{-1}\notin H$.

kinggeorge · Answer

This all in $G=S_4$.

kinggeorge · Answer

I'll be back in a few minutes to answer any remaining questions.

zzr0ck3r · Answer

nah good idea, I did not even think about permutations good idea:) ty again

kinggeorge · Answer

You're welcome.

kinggeorge · Answer

It's usually a good idea to try things with the symmetric group. Especially since every group is isomorphic to a subgroup of the symmetric group.

zzr0ck3r · Answer

hmm, that must be the great theorem that we are getting too. very n eet

zzr0ck3r · Answer

How do we know ab^-1 its not in H without producing all the elements?

kinggeorge · Answer

Well, H doesn't have many elements, so you can list the out. $H=\{e,(134),(143)\}$.

You could also notice that (134) doesn't act on 2, while (124) does. So the group generated by (134) can't act on 2 at all. Thus, (124) is not in H.

zzr0ck3r · Answer

Great I was just looking for some insight ty

kinggeorge · Answer

You're welcome.