# 剰余群の定義

$(\cdot)$: $G \times G \longrightarrow G$, $a \cdot b \longmapsto ab$
$(G, \cdot)$: a group
$N$: a normal subgroup of $G$
$G/N := \{ aN | a \in G \}$

$(\cdot)$: $G/N \times G/N \longrightarrow G/N$, $(aN) \cdot (bN) \longmapsto (aN)(bN) = \{ (an_{1})(bn_{2}) | n_{1}, n_{2} \in N \}$
This map is well-defined.
${\rm \bf Proof}$
$aN = xN, bN = yN \in G/N$
$$(aN)(bN) = a(Nb)N = a(bN)N = (ab)(NN) = (ab)N$$ and, $$(aN)(bN) = (ab)N = a(bN) = a(yN) = a(Ny) = (aN)y = (xN)y = x(Ny) = x(yN) = (xy)N = (xN)(yN)$$
So, this is an operation because $a \cdot b \longmapsto ab$ is an operation.

then $(G/N, \cdot)$ is a group, called a quotient group.
${\rm \bf Proof}$
$a$, $b$, $c\in G$ $$((aN)(bN))(cN) = ((ab)N)(cN) = ((ab)c)N = (a(bc))N = (aN)((bc)N) = (aN)((bN)(cN))$$
$a \in G$
$e$: identity element of $G$ $$(aN)(eN) = (ae)N = aN = (ea)N = (eN)(aN)$$
$a \in G$
$a^{-1}$: inverse element of $a$ $$(aN)(a^{-1}N) = (aa^{-1})N = eN = (a^{-1}a)N = (a^{-1}N)(aN)$$

$\pi: G \longrightarrow G/N$, $g \longmapsto gN$ is surjective group homomorphism. (called natural homomorphism.)
${\rm \bf Proof}$
$a$, $b \in G$ $$\pi(a)\pi(b) = (aN)(bN) = (ab)N = \pi(ab)$$