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:

[$$$\forall g \in G, \quad x \sim_{R} y \implies \quad ? \tag{1}$$$]

A similar argument shows that the equivalence relation:

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

Has the property:

[$$$\forall g \in G, \quad x \sim_{L} y \implies \quad ? \tag{2}$$$]

How are these properties related to quotient groups?