Exercise 1.2.6 (Dihedral subgroups)

Question

Let $x$ and $y$ be elements of order $2$ in any group $G$. Prove that if $t = xy$ then $tx = xt^{-1}$ (so that if $n =|xy| < \infty$ then $x,t$ satisfy the same relations in $G$ as $s,r$ in $D_{2n}$).

Richard Ganaye · Accepted Answer

Let $t = xy$, where $x,y$ are elements of $G$ of order $2$. Then $x^2 = y^2 =e$, and $t^{-1} = y^{-1} x^{-1} = yx$, so
\begin{align*}
tx &= xy x\
xt^{-1} &= x y^{-1} x^{-1} = xyx.
\end{align*}
This shows that $tx = xt^{-1}$.
\end{proof}

(If $n = |t| < \infty$, and $x 
ot \in  \langle t angle$, then $\langle x,y angle = \langle x, t angle = \{x^{i} t^j \mid 0 \leq i < 2, 0 \leq j < n\}$ is a subgroup of $G$ isomorphic to $D_{2n}$.)

Exercise 1.2.6 (Dihedral subgroups)

Answers

Comments

Exercise 1.2.6 (Dihedral subgroups)

Answers

Comments

Add answer