- Thread starter
- #1

#### buckeye1973

##### New member

- Feb 22, 2012

- 5

I am trying to understand Lagrange's Theorem by working through some exercises relating to the Orbit-Stabilizer Theorem (which I also do not fully understand.) I think essentially I'm needing to learn how to show cosets are equivalent to other things or each other.

Here's the setup:

Let $G$ be a group and $H$ be a subgroup of $G$. Define a relation on $G$ by:

$x \equiv_H y$ if and only if $x^{-1}y \in H$

I have already shown that $\equiv_H$ is an equivalence relation. Now I need to show that the $\equiv_H$ equivalence classes are the left cosets $gH$ of $H$ in $G$.

Here is my try:

Let $x,y \in G$ such that $x \equiv_H y$.

Then $x^{-1}y \in H$, which implies $(x^{-1}y)^{-1} = y^{-1}x \in H$.

So $y^{-1}x = h_y$ for some $h_y \in H$.

Then $y(y^{-1}x) = yh_y \in yH$.

thus $x = yh_y \in yH$.

Now $e \in H$, so there exists $h_x \in H$ such that $xh_x = yh_y$.

Um, Q.E.D.?

The logic flow doesn't feel right to me here. My other plan was to assume $xH \neq yH$ and show that that implies $x \not\equiv_H y$, but I couldn't even get that off the ground.

I'd really appreciate any ideas I can get here. Thanks!

Brian