# Divisibility is quotient-closed

From Groupprops

(Redirected from Divisibility is inherited by quotient groups)

## Statement

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 .

## Proof

**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.