# Another group theory problem sorry (1 Viewer)

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#### betty2301

urgent another group theory problem sorry

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
commutator

3. 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

Science Advisor
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Gold Member
Re: urgent another group theory problem sorry

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.

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