Abstract Algebra

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Quotient Group: When Can We Take $n$th Roots in a Quotient Group?

ABSALG-KXF5VI

Let $G$ be a group with neutral element $e$ and let $K$ be a normal subgroup of $G$. Let $n\ge 1$ be an integer.

Which of the following is equivalent to the property that: Every element of $G/K$ has an $n$th root in $G/K$?

A

There is a $y\in K$ such that $x=y^n$ for all $x\in G$.

B

For every $x\in G$, there is a $y\in K$ such that $x=y^n$.

C

There is a $y\in G$ such that ${xy}^{-n}\in K$ for all $x\in G$.

D

For every $x\in G$, there is a $y\in K$ such that ${xy}^n\in K$.