Exercise 1.2.16 ($D_4$ is the Klein four-group)

Question

Show that the group $\langle x_1,y_1 \mid x_1^2 = y_1^2 = (x_1y_1)^2 = 1 \rangle$ is the dihedral group $D_4$ (where $x_1$ may be replaced by the letter $r$ and $y_1$ by $s$). [Show that the last relation is the same as: $x_1y_1 = y_1 x_1^{-1}$.]

Richard Ganaye · Accepted Answer

\begin{proof} 
By Exercise 8, in the particular case $n= 2$, we have
$$\langle x_1,y_1 \mid x_1^2 = y_1^2 = (x_1y_1)^2 = 1 angle \simeq D_4.$$
Moreover, since $y_1^2 = 1$, then $y_1 = y_1^{-1}$, therefore
\begin{align*}
(x_1y_1)^2 = 1 &\iff x_1 y_1 x_1 = y_1^{-1}\
&\iff x_1 y_1 x_1 = y_1\
&\iff x_1y_1 = y_1 x_1^{-1}
\end{align*}
So
$$D_4 \simeq \langle x_1,y_1 \mid x_1^2 = y_1^2 = 1, x_1y_1 = y_1 x_1^{-1} angle,$$
which is the usual presentation of $D_4$.

($D_4$ is the Klein four-group, which is Abelian, and the order of the elements of $D_4$ distinct of $e$ is $2$.)
\end{proof}

Exercise 1.2.16 ($D_4$ is the Klein four-group)

Answers

Comments

Exercise 1.2.16 ($D_4$ is the Klein four-group)

Answers

Comments

Add answer