Is a union of cosets of a normal subgroup a subgroup itself?

Question

We have a group $G$ and $H$ is its normal subgroup. Say $K = \{C_1,...,C_k\}$ is a subgroup of $G/H$. Is the union C$_1 ∪···∪ C_k$ a subgroup of $G$?

I can see that $C_i$’s are elements of the factor group $G/H$ and thus $K$ is a union of cosets of $G$ wrt $H$. I also intuitively see that not any union of cosets is a subgroup, but I cannot deduce anything from $K$ being a subgroup though I think this fact is crucial

Answer 1

The union of the cosets is the preimage of the subgroup $K$ of $G/H$ under the quotient map $G\to G/H$, and so it is a subgroup.