Math and science::Algebra::Aluffi

Quotient group. From equivalence condition.

Under what conditions does an equivalence relation induce a quotient that is a group?

Quotient group

Let $(G, \cdot_{G})$ be a group. Let $\sim$ be an equivalence relation on the set $G$ . Let $π : G \to G / \sim$ be the quotient map induced by $\sim$ . Then $(G / \sim, \cdot_{G / \sim})$ is a group with operation:

π (a) \cdot_{G / \sim} π (b) := π (a \cdot_{G} b)

iff $π$ is [what?].

In turn, this is true of $π$ iff:

[

\begin{aligned} (1) & \forall a, a^{'}, g \in G, \\ (2) & a \sim a^{'} ⟹ ? \land ? \end{aligned}

]

04.04.2022