About solvable groups Let $L$ be a finite Galois group that's solvable. So by definition there exists a chain of normal subgroups of $L$ such that $1 = G_{0} \trianglelefteq G_{1} \trianglelefteq \dots \trianglelefteq G_{n}=L $ where all the $G_{i+1}/G_{i}$ are abelian.
Now, it's said that one can assume the $G_{i+1}/G_{i}$ to be cyclic by the structure theorem for finite abelian groups.
Why exactly is that?

A quote from wikipedia: A group $G$ is called solvable if it has a subnormal series whose factor groups are all abelian. For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a finite group has finite composition length, and every simple abelian group is cyclic of prime order. 
For more details see the proof here.
