Is G/N Abelian If N Contains All Commutators in a Group?

[betty2301]

urgent another group theory problem sorry

Homework Statement

Let G be a group with normal subgroup N. Prove that G/N is an abelian group of and only of N contains elements $aba^{-1}b^{-1}$ for all a,b in G.

Homework Equations

commutator

The Attempt at a Solution

G/N i know it is the factor group...but abelian factor group is really new to me.
my knowdge in commutator is weak as my professor did not teach this.
help!1

 

[jbunniii]

An element of the form $aba^{-1}b^{-1}$ is called a commutator. The standard notation is $[a,b] = aba^{-1}b^{-1}$.

Note that $a$ and $b$ commute iff $[a,b] = 1$.

So you need to show that $[aN, bN] = 1$ iff $[a,b] \in N$. There isn't much to it.