Exercise 3.1.40 (Commutators)

Question

Let $G$ be a group, let $N$ be a normal subgroup of $G$ and let $\overline{G} = G/N$. Prove that $\overline{x}$ and $\overline{y}$ commute in $\overline{G}$ if and only if $x^{-1} y^{-1} x y \in N$. (The element $x^{-1} y^{-1} x y $ is called the {\bf commutator} of $x$ and $y$ and is denoted by $[x,y]$.)

Richard Ganaye · Accepted Answer

\begin{proof} Let $\overline{x} = xN \in \overline{G}$ and $\overline{y} = y N \in \overline{G}$, where $x, y \in G$. Then
\begin{align*}
\overline{x}\, \overline{y} = \overline{y}\,\overline{x} & \iff \overline{x} ^{-1} \overline{y}^{-1} \overline{x}\, \overline{y} = \overline{1}\
&\iff\overline{x^{-1} y^{-1} x y } = \overline{1}\
&\iff x^{-1} y^{-1} x y N = x^{-1} y^{-1} x y \
&\iff x^{-1} y^{-1} x y \in N.
\end{align*}
So $\overline{x}$ and $\overline{y}$ commute in $\overline{G}$ if and only if $x^{-1} y^{-1} x y \in N$.

\end{proof}

Exercise 3.1.40 (Commutators)

Answers

Comments

Exercise 3.1.40 (Commutators)

Answers

Comments

Add answer