Exercise 3.4.3 ($QD_{16}$ is solvable)

Question

Find a composition series for the quasidihedral group of order $16$ (cf. Exercise 11, Section 2.5). Deduce that $QD_{16}$ is solvable.

Richard Ganaye · Accepted Answer

\begin{proof} Lattice of subgroups of $QD_{16}$:


      \bigskip    
      \begin{center}
\begin{tikzpicture}

ode (1)   at (0,0) {$\langle 1 angle$};

ode (2)   at (0,2) {$\langle \sigma^4angle$};

ode (3)   at (-1.7,2) {$\langle	au angle$};

ode (4)   at (-3.4,2) {$\langle  	au\sigma^4 angle$};

ode (5)   at (-5.1,2) {$\langle 	au \sigma^6 angle$};

ode (6)   at (-6.8,2) {$\langle 	au \sigma^2 angle$};

ode (7)    at (5,4) {$\langle 	au \sigma^7 angle$};

ode (8)    at (3,4) {$\langle 	au \sigma angle$};

ode (9)    at (0,4) {$\langle \sigma^2 angle$};

ode (10)    at (-3,4) {$\langle \sigma^4, 	au  angle$};

ode (11)    at (-6,4) {$\langle \sigma^4, 	au  \sigma^2 angle$};

ode (12)    at (3,6) {$\langle \sigma^2, 	au \sigma angle$};

ode (13)    at (0,6) {$\langle \sigma angle$};

ode (14)    at (-3,6) {$\langle \sigma^2, 	au  angle$};

ode (15)    at (0,8) {$\qquad \qquad \langle \sigma, 	au angle = QD_{16}$};

\draw (1) edge  (2) edge (3) edge (4) edge (5) edge (6);
\draw (2) edge  (7) edge (8) edge (9) edge (10) edge (11);
\draw (11) edge (5) edge (6);
\draw (10) edge (3) edge (4);
\draw (9) edge  (13);
\draw (15) edge  (13) edge (12) edge (14);
\draw (14) edge  (9) edge (10) edge (11);
\draw (12) edge  (9) edge (8) edge (7);
\end{tikzpicture}
\end{center}

A  composition series (among 10) of $QD_{16}$ is
$$\langle 1 angle \unlhd \langle \sigma^4angle \unlhd \langle \sigma^2angle \unlhd \langle \sigma angle \unlhd QD_{16}.$$
Since $|H_{i+1} : H_i | = 2$, we obtain $H_i \unlhd H_{i+1}$ $(0\leq i \leq3)$  for $H_0 = \langle 1 angle$, $H_1 = \langle \sigma^4 angle$, $H_2 = \langle \sigma^2 angle$, $H_3 = \langle \sigma angle$, $H_4 = D_8$.

Since all the composition factors are isomorphic to $Z_2$, where $Z_2$ is abelian, then $QD_{16}$ is solvable.
\end{proof}

Exercise 3.4.3 ($QD_{16}$ is solvable)

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Exercise 3.4.3 ($QD_{16}$ is solvable)

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