Definition
For group , and normal subgroup , the quotient group of by is defined as the group .
Importantly, carries group structure. This means that , , and .
For group G, and normal subgroup N⊴G, the quotient group of G by N is defined as the group (G/N,⋅).
Importantly, G/N carries group structure. This means that (gN)∙(g′N)=(g⋅g′)N, eG/N=eGN=N, and (gN)−1=g−1N.