factor group

**GTK X Hunter** · October 21st 2009, 10:47 AM

Let G be any group, and $N \triangleleft G$ . Show that $aba^{-1}b^{-1}\in N, \forall a,b\in N \Longleftrightarrow G/N$ is abelian

**Swlabr** · October 21st 2009, 11:11 AM

Originally Posted by GTK X Hunter

Let G be any group, and $N \triangleleft G$ . Show that $aba^{-1}b^{-1}\in N, \forall a,b\in N \Longleftrightarrow G/N$ is abelian

Hint: Prove that $aba^{-1}b^{-1}=1 \forall a,b \in G \Leftrightarrow G$ is abelian.

**GTK X Hunter** · March 27th 2010, 12:32 AM

Originally Posted by Swlabr

Hint: Prove that $aba^{-1}b^{-1}=1 \forall a,b \in G \Leftrightarrow G$ is abelian.

do you mean $aba^{-1}b^{-1}=e$
what is the reason?

**Swlabr** · March 27th 2010, 07:44 AM

Originally Posted by GTK X Hunter

do you mean $aba^{-1}b^{-1}=e$
what is the reason?

1 and e are both commonly used to denote the identity element.

What does this equality tell you? By definition, $ab=ba \: \forall a, b \in G \Leftrightarrow G$ is abelian. Can you get this from the equation I gave you above?

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