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