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.