Divisibility is quotient-closed
Suppose is a group and is a normal subgroup of , so that is the quotient group. Then, if is a natural number such that is -divisible (i.e., every element of has a root in , then so is , i.e., every element of has a root in .
Given: A group and a natural number such that every element of has a root in , a normal subgroup with quotient group . An element .
To prove: There exists such that .
Proof: Let be the quotient map and let be such that . Since is -divisible, there exists such that . Let . Then, , so is as desired.