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

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SUMMARY

The discussion centers on proving that the quotient group G/N is abelian if and only if the normal subgroup N contains all commutators of the form aba^{-1}b^{-1} for all elements a, b in the group G. A commutator is denoted as [a, b] = aba^{-1}b^{-1}, and the condition for G/N to be abelian is that [aN, bN] = 1 if and only if [a, b] is an element of N. This establishes a clear relationship between the structure of the group and the properties of its normal subgroup.

PREREQUISITES
  • Understanding of group theory concepts, particularly normal subgroups.
  • Familiarity with commutators and their notation in group theory.
  • Knowledge of quotient groups and their properties.
  • Basic proficiency in algebraic manipulation within group structures.
NEXT STEPS
  • Study the properties of normal subgroups in group theory.
  • Learn about commutators and their significance in group structures.
  • Research the concept of quotient groups and their applications.
  • Explore examples of abelian groups and the conditions for their formation.
USEFUL FOR

This discussion is beneficial for students of abstract algebra, particularly those studying group theory, as well as educators seeking to clarify the relationship between normal subgroups and commutators in group structures.

betty2301
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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
 
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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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