Sundry Coset Results

Theorem 1

 * $$x \in y H \iff x^{-1} y \in H$$;
 * $$x \in H y \iff x y^{-1} \in H$$.

Thus:


 * $$x, y$$ are in the same left coset of $$H$$ iff $$x^{-1} y \in H$$;
 * $$x, y$$ are in the same right coset of $$H$$ iff $$x y^{-1} \in H$$.

Theorem 2

 * $$x H = y H \iff x^{-1} y \in H$$;
 * $$H x = H y \iff x y^{-1} \in H$$.

Theorem 3

 * $$x H = H \iff x \in H$$;
 * $$H x = H \iff x \in H$$.

Theorem 4

 * The mapping $$\lambda_x: H \to x H$$, where $$\lambda_x$$ is the left regular representation of $$H$$ with respect to $$x$$, is a bijection from $$H$$ to $$x H$$.


 * The mapping $$\rho_x: H \to H x$$, where $$\rho_x$$ is the right regular representation of $$H$$ with respect to $$x$$, is a bijection from $$H$$ to $$H x$$.

Proof of Theorem 1

 * $$x \in y H \iff x^{-1} y \in H$$:

$$ $$ $$ $$ $$


 * $$x \in H y \iff x y^{-1} \in H$$ is proved similarly.

Proof of Theorem 2

 * $$x H = y H \iff x^{-1} y \in H$$:

$$ $$ $$


 * $$H x = H y \iff x y^{-1} \in H$$ is proved similarly.

Proof of Theorem 3

 * $$x H = H \iff x \in H$$:

$$ $$ $$ $$


 * $$H x = H \iff x \in H$$ is proved similarly.

Proof of Theorem 4
Follows from Regular Representations in Group are Permutations.