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

$\pi(a) \groupMul{G/\!\sim} \pi(b) := \pi( a \groupMul{G} b)$

iff $$\pi$$ is [what?].

In turn, this is true of $$\pi$$ iff:

[\begin{align} \forall a, a', g \in G, & \\ & a \sim a' \implies \quad ? \quad \land \quad ? \; \end{align}]