Another group theory problem sorry (1 Viewer)

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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 [itex]aba^{-1}b^{-1}[/itex] for all a,b in G.

2. Relevant equations

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.


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Re: urgent another group theory problem sorry

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

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

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

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