# Prove G/N is abelian if and only if aba^-1b^-1 in N for all a,b in G.?

### 1 Answer

Relevance

- spoon737Lv 61 decade agoFavorite Answer
Suppose G/N is Abelian. Then,

abN = (aN)(bN) = (bN)(aN) = baN

So, N = (ab)(ba)^-1N = aba^-1b^-1N, which means aba^-1b^-1 is in N.

Conversely, suppose aba^-1b^-1 is in N. Then,

(bN)(aN)

= baN

= Nba

=Naba^-1b^-1ba

=Nab

=abN

=(aN)(bN)

Thus, G/N is Abelian.

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