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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:

\[ x \sim_{R} y \iff xy^{-1} \in H \]

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

Or a (possibly different) equivalence relation:

\[ x \sim_{L} y \iff x^{-1}y \in H \]

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

Property

The equivalence relation:

\[ x \sim_{R} y \iff xy^{-1} \in H \]

Has the following important property:

\[ \begin{equation} \forall g \in G, \quad x \sim_{R} y \implies xg \sim yg \tag{1} \end{equation} \]

A similar argument shows that the equivalence relation:

\[ x \sim_{L} y \iff x^{-1}y \in H \]

Has the property:

\[ \begin{equation} \forall g \in G, \quad x \sim_{L} y \implies gx \sim gy \tag{2} \end{equation} \]

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

\( xy^{-1} \in H \) implies that \( x \) is in the coset \( Hy \) and that \( x \) can be expressed as \( h_1 y \) for some \( h_1 \in H \). Then, we see that for \( x \) we have \( xg = h_1 y g \) and for \( y \) we have \( yg = e y g \). Thus, both \( xg \) and \( yg \) are in the same right coset of \( H \), which is the requirement for equivalence.

Proof for right cosets (v2)

Proof.

\[ \begin{align*} xy^{-1} \in H &\implies x(gg^{-1})y^{-1} \in H \\ &\implies (xg)(g^{-1}y^{-1}) \in H \\ &\implies (xg)(yg)^{-1} \in H \\ \end{align*} \]

So, we have:

\[ x \sim_{R} y \implies xg \sim_{R} yg \]

Property 1. Visualization.