Showing that Equivalence Relations are the Same.

[jmjlt88]

Let G be a group and let H be a subgroup of G.

Define ~ as a~b iff ab^-1εH.

Define ~~ as a~~b iff a^-1bεH.

The book I am using wanted us to prove that each was an equivalence relation, which was easy. Then, it asked if these equivalence relations were the same, if so, prove it. My initial reaction was yes. I did not prove it, but I did write down a quick idea surronded by question marks and "ask PhysicsForum!." Now that I know a bit more about cosets, I say no.

For my idea, I wrote something like this. [Remember, I am writing to me.]

"Show that a~b implies a~~b and vice versa. If a~b, then ab^-1εH. Show that this implies that a and b^-1 are in H... then a^-1 and b are in H. Hence, a^-1bεH and a~~b... Similar going the other way... But, not sure if this even works ? Ask PhysicsForum before trying to write this out."

Well, any help?? :)

 

[micromass]

Try to find a counterexample.

I claim that it is true for normal subgroups, can you prove that?

So to find a counterexample, pick your favorite nonnormal subgroup and do something with it.

 

[jmjlt88]

Thanks! :)