If N is a normal subgroup of a group G, then prove that G/N is a group with respect to multiplication of cosets.

Question

I can prove that G/N follows closure property, associative property, has identity and inverse with respect to multiplication of cosets. But after this, what should I do to prove that G/N is a group with respect to multiplication of cosets?

Answer 1

Let $\;1\cdot N=N\;$ be an element in $\;G/N\;$ , with $\;1\;$ the unit in the group $\;G\;$ , then for any $\;g\in G\;$ we get

$$(gN)(1N):=(g\cdot 1)N=gN\implies N\;\text{ is the unit in the group}\;\;G/N$$