Math and science::Algebra::Aluffi

Quotient groups. From Cosets.

This card connects cosets and quotients.

Coset

For any subgroup $H$ of $G$ , we can form an equivalence relation:

x \sim_{R} y ⟺ x y^{- 1} \in H

The partition created by this equivalence relation is the set of right cosets.

Or a (possibly different) equivalence relation:

x \sim_{L} y ⟺ x^{- 1} y \in H

The partition created by this equivalence relation is the set of left cosets.

Property

The equivalence relation:

x \sim_{R} y ⟺ x y^{- 1} \in H

Has the following important property:

\begin{matrix} (1) & \forall g \in G, x \sim_{R} y ⟹ x g \sim y g \end{matrix}

A similar argument shows that the equivalence relation:

x \sim_{L} y ⟺ x^{- 1} y \in H

Has the property:

\begin{matrix} (2) & \forall g \in G, x \sim_{L} y ⟹ g x \sim g y \end{matrix}

How are these properties related to quotient groups?

Connection to quotient groups

An equivalence relation that satisfies (1) and (2) forms a quotient that is a group, whose operation is inherited from the original group.

Proof for right cosets

$x y^{- 1} \in H$ implies that $x$ is in the coset $H y$ and that $x$ can be expressed as $h_{1} y$ for some $h_{1} \in H$ . Then, we see that for $x$ we have $x g = h_{1} y g$ and for $y$ we have $y g = e y g$ . Thus, both $x g$ and $y g$ are in the same right coset of $H$ , which is the requirement for equivalence.

Proof for right cosets (v2)

Proof.

\begin{aligned} x y^{- 1} \in H & ⟹ x (g g^{- 1}) y^{- 1} \in H \\ ⟹ (x g) (g^{- 1} y^{- 1}) \in H \\ ⟹ (x g) (y g)^{- 1} \in H \end{aligned}

So, we have:

x \sim_{R} y ⟹ x g \sim_{R} y g

Property 1. Visualization.

Some extra visualizations:

and