Originally Posted by
Giraffro Let $\displaystyle D_{2n} = \langle \sigma, \tau \mid \sigma^4 = e = \tau^2, \tau^{-1} \sigma \tau = \sigma^{-1} \rangle$
Correcting a possible minor typo: it must be $\displaystyle \sigma^n=1\,,\,\,not\,\,\,\sigma^4=1$.
Tonio
be a dihedral group of order $\displaystyle 2n$. Then $\displaystyle [D_{2n} : \langle \sigma \rangle] = 2$, so $\displaystyle \langle \sigma \rangle \lhd D_{2n}$. Therefore $\displaystyle \{e\} \lhd \langle \sigma \rangle \lhd D_{2n}$ is a subnormal series of $\displaystyle D_{2n}$ with Abelian factors, so $\displaystyle D_{2n}$ is solvable.