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:
[\[
\begin{equation}
\forall g \in G, \quad x \sim_{R} y \implies \quad ? \tag{1}
\end{equation}
\]]
A similar argument shows that the equivalence relation:
\[ x \sim_{L} y \iff x^{-1}y \in H \]
Has the property:
[\[
\begin{equation}
\forall g \in G, \quad x \sim_{L} y \implies \quad ? \tag{2}
\end{equation}
\]]
How are these properties related to quotient groups?