I actually draw the question from (2) of this problem.
I quote it here：
Let $N$ be a normal subgroup of a group $G$ of index $4$. Show
(1) that $G$ contains a subgroup of index $2$,
(2) that if $G/N$ is not cyclic, then there exists three proper normal subgroups $A,B$ and $C$ of $G$ such that $G=A \cup B \cup C$.
When $G/N$ is not cyclic, it must be Klein four-group. I know that then $G/N$ can be a union of three proper subgroups of $G/N$; however, the hint also implies that “thereby $G$ is definitely union of three proper subgroups of $G$”. I can pretty accept it, but feel quite confused when trying to sort it out.
1) How to make it clear to understand? In both senses of strict proof and reasonable thinking.
2) Furthermore, I feel very strongly that quotient groups are just like “contracted groups having the same shape with the original”. But it’s just my imagination, where can I get specific instructions?
Any help is sincerely appreciated.