Math and science::Algebra::Aluffi

Quotient groups. From Cosets.

This card connects cosets and quotients.

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.

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 ⟹ ? \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 ⟹ ? \end{matrix}

]

How are these properties related to quotient groups?

04.04.2022