User:Dfeuer/Coset stuff in progress

Theorem

Let $\left({G, \circ}\right)$ be a group and let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let:

• $y \circ H$ denote the left coset of $H$ by $y$
• $H \circ y$ denote the right coset of $H$ by $y$.

Then:

\(\text {(1)}: \quad\)

\(\ds x \in y \circ H\)

\(\iff\)

\(\ds x^{-1} \circ y \in H\)

\(\text {(2)}: \quad\)

\(\ds x \in H \circ y\)

\(\iff\)

\(\ds x \circ y^{-1} \in H\)

Theorem

Let $\left({G, \circ}\right)$ be a group and let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let:

• $y \circ H$ denote the left coset of $H$ by $y$
• $H \circ y$ denote the right coset of $H$ by $y$.

Then:

\(\text {(1)}: \quad\)

\(\ds x \in y \circ H\)

\(\iff\)

\(\ds x^{-1} \circ y \in H\)

Proof

\(\ds \)

\(\)

\(\ds x \in y H\)

\(\ds \)

\(\iff\)

\(\ds \exists h' \in H: x = y h'\)

Definition of Left Coset

\(\ds \)

\(\iff\)

\(\ds \exists h = h'^{-1} \in H: x h = y\)

Group element properties

\(\ds \)

\(\iff\)

\(\ds \exists h \in H: h = x^{-1} y\)

Group element properties

\(\ds \)

\(\iff\)

\(\ds x^{-1} y \in H\)

Definition of Left Coset

$\blacksquare$

Theorem

Let $\left({G, \circ}\right)$ be a group and let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let:

• $y \circ H$ denote the left coset of $H$ by $y$
• $H \circ y$ denote the right coset of $H$ by $y$.

Then:

\(\text {(1)}: \quad\)

\(\ds x \in H \circ y\)

\(\iff\)

\(\ds x \circ y^{-1} \in H\)

Proof

Let $\left({G,*}\right)$ be the opposite group of $\left({G,\circ}\right)$.

Then:

$x \in H \circ y \iff x \in y * H$

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

Since $H$ is closed under inverses:

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

By Element in Left Coset iff Product with Inverse in Subgroup:

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

Thus $(2)$ holds.

$\blacksquare$